THE 


MECHANICAL   PRINCIPLES 


ENGINES  KING 


ARCHITECTURE. 


HENRY  MOSELET,  I. A.  F.R.S. 

, 

CHAPLAIN    IN    ORDINARY  TO  THE    QUEEN,  CANON    OF  BRISTOL,  VICAR   OF  OLVESTON  ; 

CORRESPONDING   MEMBER  OF   TUG   INSTITUTE   OF   FRANCE,  AND   FORMERLY   PROFESSOB 
OF   NATURAL   PHILOSOPHY   AND   ASTRONOMY   IN   KING'S   COLLEGE,  LONDON. 


Second  American  from  Second  London  Edition 
WITH    ADDITIONS    BY 

D.    H.    M  A  H  A  N  ,    LL.D. 

U.    S.    MILITARY   ACADEMY. 

V.MTIJ     ri,  LUSTRATIONS    ON    WOOD. 


NEW  YORK  : 

JOHN  WILEY  &  SON,  535  BROADWAY 
1 


ENTERFO  according  *o  Act  of  Congress,  <jt\  the  year  1856,  Ij1 
WILEY    &    HA-LSTRD, 

fc  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the  Southu  a  District 

of  New  York. 


EDITOR'S     PREFACE. 


THE  high  place  that  Professor  Moseley  occupies  in  the 
scientific  world,  as  an  original  investigator,  and  the  clear- 
ness and  elegance  of  the  methods  he  has  employed  in  this 
work  have  made  it  a  standard  text  book  on  the  subjects  it 
treats  of.  In  undertaking  its  revision  for  the  press,  at  the 
request  of  the  publishers  of  this  edition,  it  has  been  deemed 
advisable,  in  view  of  the  class  of  students  into  whose  hands 
it  may  fall,  to  make  some  slight  addition  to  the  original. 
This  has  been  done  in  the  way  of  Notes  thrown  into  an 
Appendix,  the  matter  of  which  has  been  gathered  from 
various  authorities  ;  but  chiefly  from  notes  taken  by  the 
editor,  whilst  a  pupil  at  the  French  military  school  at  Metz, 
of  lectures  delivered  by  General  Poncelet,  at  that  time,  1829, 
professor  in  that  school.  It  is  a  source  of  great  pleasure  to 
the  editor  to  have  this  opportunity  of  publicly  acknowledg- 
ing his  obligations  to  the  teachings  of  this  eminent  savan, 
who  is  distinguished  not  more  for  his  high  scientific  attain- 
ment, and  the  advancement  he  has  given  to  mechanical 
science,  than  for  having  brought  these  to  minister  to  the 
wants  of  the  industrial  classes,  the  intelligent  success  of 
whose  operations  depends  so  much  upon  mechanical  science, 
by  presenting  it  in  a  form  to  render  it  attainable  by  the  most 
ordinary  capacities. 

Hi 


iv  EDITOR'S  PREFACE. 

The  editor  would  remark  that  lie  has  carefully  refrained 
from  making  any  alterations  in  the  text  revised,  except  cor- 
rections of  typographical  errors,  and  in  one  instance  where, 
from  a  repetition  of  apparently  one  of  these,  he  apprehended 
some  difficulty  might  be  offered  to  the  student  if  allowed 
to  remain  exactly  as  printed  in  the  original. 

UNITED  STATES  MILITARY  ACADEMY, 

Went  Point  March  8,  1866. 


PKEFACE  TO  THE  SECOND  EDITION. 


I  HAVE  added  in  this  Edition  articles : — first,  "  On  the 
Dynamical  Stability  of  Floating  Bodies ;"  secondly,  "  On 
the  Kolling  of  a  Cylinder  ;"  thirdly,  "  On  the  descent  of  a 
body  upon  an  inclined  plane,  when  subjected  to  variations  of 
temperature,  which  would  otherwise  rest  upon  it ;"  fourthly, 
u  On  the  state  bordering  upon  motion  of  a  body  moveable 
about  a  cylindrical  axis  of  finite  dimensions,  when  acted 
upon  by  any  number  of  pressures." 

The  conditions  of  the  dynamical  stability  of  floating 
bodies  include  those  of  the  rolling  and  pitching  motion  of 
ships.  The  discussion  of  the  rolling  motion  of  a  cylinder 
includes  that  of  the  rocking  motion  to  which  a  locomotive 
engine  is  subject,  when  its  driving  wheels  are  falsely 
balanced,  and  that  of  the  slip  of  the  wheel  due  to  the  same 
cause.  The  descent  of  a  body  upon  an  inclined  plane 
when  subjected  to  variations  in  temperature,  which  other- 
wise would  rest  upon  it,  appears  to  explain  satisfactorily  the 
descent  of  glaciers. 

The  numerous  corrections  made  in  the  text,  I  owe  chiefly 
to  my  old  pupils  at  King's  College,  to  whom  the  lectures 
of  which  it  contains  the  substance,  were  addressed.  For 


VI  PREFACE   TO   THE    SECOND    EDITION. 

several  important  ones  I  am,  however,  indebted  to  Mr 
Eobinson,  Master  of  the  School  for  Shipwrights'  Apprentices, 
in  Her  Majesty's  Dockyard,  Portsea  ;  to  whom  I  have  also  to 
express  my  warm  acknowledgments  for  the  care  with 
which  he  has  corrected  the  proof  sheets  whilst  going  through 
the  press. 

May,  1855 


PREFACE. 


IN  the  following  work,  I  have  proposed  to  myself  to  apply: 
the  principles  of  mechanics  to  the  discussion  of  the  most 
important  and  obvious  of  those  questions  which  present 
themselves  in  the  practice  of  the  engineer  and  the  architect ; 
and  I  have  sought  to  include  in  that  discussion  all  the 
circumstances  on  which  the  practical  solution  of  such  ques- 
tions may  be  assumed  to  depend.  It  includes  the  substance 
of  a  course  of  lectures  delivered,  to  >  the  students  of  King's 
College  in  the  department  of  engineering  and  architecture, 
during  the  years  1840,  1841,  1842.* 

In  the  first  part  I  have  treated  of  those  portions  of  the 
science  of  STATICS,  which  have  their  application  in  the  theory 
of  machines  and  the  theory  of  construction. 

In  the  second,  of  the  science  of  DYNAMICS,  and,  under  this 
head,  particularly  of  that  union  of  a  continued  pressure  with 
a  continued  motion  which  has  received  from  English  writers 
the  various  names  of  "dynamical  effect,"  "efficiency,"  "work 
done,"  "labouring  force,"  "work,"  &c. ;  and  "moment 
d'activite","  "quantite  d' action,"  "puissance  mecanique," 
"  travail,"  from  French  writers. 

Among  the  latter  this  variety  of  terms  has  at  length  given 
place  to  the  most  intelligible  and  the  simplest  of  them,, 

*  The  first  170  pages  of  the  work  were  printed  for  the  use  of  my  pupils  in  the- 
year  1840.  Copies  of  them  were  about  the  same  time  in  the  possession  of 
several  of  my  friends  in  the  Universities. 


Vlll  PREFACE. 

"  travail."  The  English  word  "  work  "  is  the  obvious  trans- 
lation of  "  travail,"  and  the  use  of  it  appears  to  be  recom- 
mended by  the  same  considerations.  The  work  of  overcoming 
a  pressure  of  one  pound  through  a  space  of  one  foot  has,  in 
this  country,  been  taken  as  the  unit,  in  terms  of  which  any 
other  amount  of  work  is  estimated ;  and  in  France,  the  work 
of  overcoming  a  pressure  of  one  kilogramme  through  a  space 
of  one  metre.  M.  Dupiii  has  proposed  the  application  of  the 
term  dyname  to  this  unit. 

I  have  gladly  sheltered  myself  from  the  charge  of  having 
contributed  to  increase  the  vocabulary  of  scientific  words, 
by  assuming  the  obvious  term  "  unit  of  work  "  to  represent 
concisely  and  conveniently  enough  the  idea  which  is  attached 
to  it. 

The  work  of  any  pressure  operating  through  any  space  is 
evidently  measured  in  terms  of  such  units,  oy  multiplying 
the  number  of  pounds  in  the  pressure  by  the  number  of  feet 
in  the  space,  if  the  direction  of  the  pressure  be  continually 
that  in  which  the  space  is  described.  If  not,  it  follows,  by 
a  simple  geometrical  deduction,  that  it  is  measured  by  the 
product  of  the  number  of  pounds  in  the  pressure,  by  the 
number  of  feet  in  the  projection  of  the  space  described,* 
upon  the  direction  of  the  pressure ;  that  is,  by  the  product 
of  the  pressure  by  its  virtual  velocity.  Thus,  then,  we 
conclude  at  once,  by  the  principle  of  virtual  velocities,  that 
if  a  machine  work  under  a  constant  equilibrium  of  the 
pressures  applied  to.  it,  or  if  it  work  uniformly,  then  is  the 
aggregate  work  of  those  pressures  which  tend  to  accelerate 
its  motion  equal  to  the  aggregate  work  of  those  which  tend 
to  retard  it ;  and,  by  the  principle  of  vis  viva,  that  if  the 
machine  do  not  work  under  an  equilibrium  of  the  forces 
impressed  upon  it,  then  is  the  aggregate  work  of  those  which 
tend  to  accelerate  the  motion  of  the  machine  greater  or  less 

*  If  the  direction  of  the  pressure  renfain  always  parallel  to  itself,  the  space 
described  may  be  any  finite  space  ;  if  it  do  not,  the  space  is  understood  to  be 
so  small,  that  the  direction  of  the  pressure  may  be  supposed  to  remain  parallel 
to  itself  whilst  that  space  is  described. 


PREFACE.  IX 

than  the  aggregate  work  of  those  which  tend  to  retard  its 
motion  by  one  half  the  aggregate  of  the  vires  vivce  acquired 
or  lost  by  the  moving  parts  of  the  system,  whilst  the  work  is 
being  done  upon  it.  In  no  respect  have  the  labours  of  the 
illustrious  president  of  the  Academy  of  Sciences  more  con- 
tributed to  the  development  of  the  theory  of  machines  than 
in  the  application  which  he  has  so  successfully  made  to  it  of 
this  principle  of  vis  viva.*  In  the  elementary  discussion  of 
this  principle,  which  is  given  by  M.  Poncelet,  in  the  intro- 
duction to  his  Mecanique  Industrielle,  he  has  revived  the 
term  vis  inertia  (vis  inertias,  vis  insita,  Newton),  and, 
associating  with  it  the  definitive  idea  of  a  force  of  resistance 
opposed  to  the  acceleration  or  the  retardation  of  a  body's 
motion,  he  has  shown  (Arts.  66.  and  122.)  the  work  expended 
in  overcoming  this  resistance  through  any  space,  to  be 
measured  by  one  half  the  vis  viva  accumulated  through  the 
space ;  so  that  throwing  into  the  consideration  of  the  forces 
under  which  a  machine  works,  the  vires  inerticB  of  its  moving 
elements,  and  observing  that  one  half  of  their  aggregate  vis 
viva  is  equal  to  the  aggregate  work  of  their  vires  inertice,  it 
follows,  by  the  principle  of  virtual  velocities,  that  the  differ- 
ence between  the  aggregate  work  of  those  forces  impressed 
upon  a  machine,  which  tend  to  accelerate  its  motion,  and 
the  aggregate  work  of  those  which  tend  to  retard  the  motion, 
is  equal  to  the  aggregate  work  of  the  vires  inerticB  of  the 
moving  parts  of  the  machine :  under  which  form  the  prin- 
ciple of  vis  viva  resolves  itself  into  the  principle  of  virtual 
velocities.  So  many  difficulties,  however,  oppose  themselves 
to  the  introduction  of  the  term  vis  inertice,  associated  with 
the  definitive  idea  of  a  force,  into  the  discussion  of  questions 
of  mechanics,  and  especially  of  practical  and  elementary 
mechanics,  that  I  have  thought  it  desirable  to  avoid  it.  It 
is  with  this  view  that  I  have  given  a  new  interpretation  to 
that  function  of  the  velocity  of  a  moving  body  which  is 
known  as  its  vis  viva.  One  half  that  function  I  have  inter- 
preted to  represent  the  number  of  units  of  work  accumulated 

*  See  Poncelet,  Mecanique  Industrielle,  troisieme  partie. 


PREFACE. 


in  the  body  so  long  as  its  motion  is  continued.  This  number 
of  units  of  work  it  is  capable  of  reproducing  upon  any  resist- 
ance opposed  to  its  motion.  A  very  simple  investigation 
(Art.  66.)  establishes  the  truth  of  this  interpretation,  and 
gives  to  the  principle  of  vis  viva  the  following  more  simple 
enunciation : — "  The  difference  between  the  aggregate  work 
done  upon  the  machine,  during  any  time,  by  those  forces 
which  tend  to  accelerate  the  motion,  and  the  aggregate 
work,  during  the  same  time,  of  those  which  tend  to  retard 
the  motion,  is  equal  to  the  aggregate  number  of  units  of 
work  accumulated  in  the  moving  parts  of  the  machine 
during  that  time  if  the  former  aggregate  exceed  the  latter, 
and  lost  from  them  during  that  time  if  the  former  aggregate 
fall  short  of  the  latter."  Tims,  then,  if  the  aggregate  work 
of  the  forces  which  tend  to  accelerate  the  motion  of  a 
machine  exceeds  that  of  the  forces  which  tend  to  retard  it, 
then  is  the  surplus  work  (that  done  upon  the  driving  points, 
above  that  expended  upon  the  prejudicial  resistances  and 
upon  the  working  points)  continually  accumulated  in  the 
moving  elements  of  the  machine,  and  their  motion  is  thereby 
continually  accelerated.  And  if  the  former  aggregate  be 
less  than  the  latter,  then  is  the  deficiency  supplied  from  the 
work  already  accumulated  in  the  moving  elements,  so  that 
their  motion  is  in  this  case  continually  retarded. 

The  moving  power  divides  itself  whilst  it  operates  in  a 
machine,  first,  into  that  which  overcomes  the  prejudicial 
resistances  of  the  machine,  or  those  which  are  opposed  by 
friction  and  other  causes,  uselessly  absorbing  the  work  in  its 
transmission.  Secondly,  into  that  which  accelerates  the 
motion  of  the  various  moving  parts  of  the  machine,  and  which 
accumulates  in  them  so  long  as  the  work  done  by  the  moving 
power  upon  it  exceeds  that  expended  upon  the  various 
resistances  opposed  to  the  motion  of  the  machine.  Thirdly, 
into  that  which  overcomes  the  useful  resistances,  or  those 
which  are  opposed  to  the  motion  of  the  machine  at  the 
working  point,  or  points,  by  the  useful  work  which  is  done 
by  it. 


PREFACE.  XI 

Between  these  three  elements  there  obtains  in  every 
machine  a  mathematical  relation,  which  I  have  called  its 
MODULUS.  The  general  form  of  this  modulus  I  have  discussed 
in  a  memoir  on  the  "  Theory  of  Machines  "  published  in  the 
Philosophical  Transactions  for  the  year  1841.  The  deter- 
mination of  the  particular  moduli  of  those  elements  of 
machinery  which  are  most  commonly  in  use,  is  the  subject 
of  the  third  part  of  the  following  work.  From  a  combination 
of  the  moduli  of  any  such  elements  there  results  at  once  the 
modulus  of  the  machine  compounded  of  them." 

"When  a  machine  has  acquired  a  state  of  uniform  motion, 
work  ceases  to  accumulate  in  its  moving  elements,  and  its 
modulus  assumes  the  form  of  a  direct  relation  between  the 
work  done  by  the  motive  power  upon  its  driving  point  and 
that  yielded  at  its  working  points.  I  have  determined  by  a 
general  method'35'  the  modulus  in  this  case,  from  that  statical 
relation  between  the  driving  and  working  pressures  upon 
the  machine  which  obtains  in  the  sfate  bordering  upon  its 
motion,  and  which  may  be  deduced  from  the  known  condi- 
tions of  equilibrium  and  the  established  laws  of  friction.  In 
making  this  deduction  I  have,  in  every  case,  availed  myself 
of  the  following  principle,  first  published  in  my  paper  on  the 
theory  of  the  arch,  read  before  the  Cambridge  Philosophical 
Society  in  Dec.  1833,  and  printed  in  their  Transactions  of 
the  following  year: — "In  the  state  bordering  upon  motion 
of  one  body  upon  the  surface  of  another,  the  resultant 
pressure  upon  their  common  surface  of  contact  is  inclined 
to  the  normal,  at  an  angle  whose  tangent  is  equal  to  the 
coefficient  of  friction." 

This  angle  I  have  called  the  limiting  angle  of  resistance. 
Its  values  calculated,  in  respect  to  a  great  variety  of  surfaces 
of  contact,  are  given  in  a  table  at  the  conclusion  of  the 
second  part,  from  the  admirable  experiments  of  M.  Morin,f 
into  the  mechanical  details  of  which  precautions  have  been 
introduced  hitherto  unknown  to  experiments  of  this  class, 

*  Art.  152.     See  Phil.  Trans.,  1841,  p.  290. 

f  Nouvelles  Experiences  sur  le  Frottement,  Paris,  1833. 


Xll  PEEFACE. 

and  which  have  given  to  our  knowledge  of  the  laws  of 
friction  a  precision  and  a  certainty  hitherto  unhoped  for. 

Of  the  various  elements  of  machinery  those  which  rotate 
about  cylindrical  axes  are  of  the  most  frequent  occurrence 
and  the  most  useful  application;  I  have,  therefore,  in  the 
first  place  sought  to  establish  the  general  relation  of  the 
state  bordering  upon  motion  between  the  driving  and  the 
working  pressures  upon  such  a  machine,  reference  being 
had  to  the  weight  of  the  machine.*  This  relation  points  out 
the  existence 'of  a  particular  direction  in  which  the  driving 
pressure  should  be  applied  to  any  such  machine,  that  the 
amount  of  work  expended  upon  the  friction  of  the  axis  may 
be  the  least  possible.  This  direction  of  the  driving  pressure 
always  presents  itself  on  the  same  side  of  the  axis  with  that 
of  the  working  pressure,  and  when  the  latter  is  vertical  it 
becomes  parallel  to  it ;  a  principle  of  the  economy  of  power 
in  machinery  which  has  received  its  application  in  the 
parallel  motion  of  the  marine  engines  known  as  the  Gorgon 
Engines. 

I  have  devoted  a  considerable  space  in  this  portion  of  my 
work  to  the  determination  of  the  modulus  of  a  system  of 
toothed  wheels ;  this  determination  I  have,  moreover, 
extended  to  bevil  wheels,  and  have  included  in  it,  with  the 
influence  of  the  friction  of  the  teeth  of  the  wheels,  that  of 
their  axes  and  their  weights.  An  approximate  form  of  this 
modulus  applies  to  any  shape  of  the  teeth  under  which  they 
may  be  made  to  work  correctly ;  and  when  in  this  approxi- 
mate form  of  the  modulus  the  terms  which  represent  the 
influence  of  the  friction  of  the  axis  and  the  weight  of  the 
wheel  are  neglected,  it  resolves  itself  into  a  well  known 
theorem  of  M.  Poncelet,  reproduced  by  M.  ISTavier  and  the 
Rev.  Dr.  Whewell.f  In  respect  to  wheels  having  epicy- 

*  In  my  memoir  on  the  "  Theory  of  Machines  "  (Phil.  Trans.  1841),  I  have 
extended  this  relation  to  the  case  in  which  the  number  of  the  pressures  and 
their  directions  are  any  whatever.  The  theorem  which  expresses  it  is  given  in 
the  Appendix  of  this  work. 

f  In  the  discussion  of  the  friction  of  the  teeth  of  wheels,  the  direction  of  the 
mutual  pressures  of  the  teeth  is  determined  by  a  method  first  applied  by  me  to 


PREFACE.  •  xiij 

cloidal  and  involute  teeth,  the  modulus  assumes  a  character 
of  mathematical  exactitude  and  precision,  and  at  once 
establishes  the  conclusion  (so  often  disputed)  that  the  loss  of 
power  is  greater  before  the  teeth  pass  the  line  of  centres 
than  at  corresponding  points  afterwards ;  that  the  contact 
should,  nevertheless,  in  all  cases  take  place  partly  before 
and  partly  after  the  line  of  centres  has  been  passed.  In  the 
case  of  involute  teeth,  the  proportion  in  which  the  arc  of 
contact  should  thus  be  divided  by  the  line  of  centres  is 
determined  by  a  simple  formula ;  as  also  are  the  best 
dimensions  of  the  base  of  the  involute,  with  a  view  to  the 
most  perfect  economy  of  power  in  the  working  of  the 
wheels. 

The  greater  portion  of  the  discussions  in  the  third  part  of 
my  work  I  believe  to  be  new  to  science.  In  the  fourth  part 
I  have  treated  of  "  the  theory  of  the  stability  of  structures," 
referring  its  conditions,  so  far  as  they  are  dependent  upon 
the  rotation  of  the  parts  of  a  structure  upon  one  another,  to 
the  properties  of  a  certain  line  which  may  be  conceived  to 
traverse  every  structure,  passing  through  those  points  in  it 
where  its  surfaces  of  contact  are  intersected  by  the  resultant 
pressures  upon  them.  To  this  line,  whose  properties  I  first 
discussed  in  a  memoir  upon  "  the  Stability  of  a  System  of 
Bodies  in  Contact,"  printed  in  the  sixth  volume  of  the  Carrib. 
Phil.  Trans.,  I  have  given  the  name  of  the  line  of  resist- 
ance ;  it  differs  essentially  in  its  properties  from  a  line 
referred  to  by  preceding  writers  under  the  name  of  the 
curve  of  equilibrium  or  the  line  of  pressure. 

The  distance  of  the  line  of  resistance  from  the  extrados  of 
a  structure,  at  the  point  where  it  most  nearly  approaches  it, 
I  have  taken  as  a  measure  of  the  stability  of  a  structure,*  and 

that  purpose  in  a  popular  treatise,  entitled  Mechanics  applied  to  the  Arts, 
published  in  1834. 

*  This  idea  was  suggested  to  me  by  a  rule  for  the  stability  of  revetement 
walls  attributed  to  Vauban,  to  the  effect,  that  the  resultant  pressure  should 
intersect  the  base  of  such  a  wall  at  a  point  whose  distance  from  its  extrados  is 
iths  the  distance  between  the  extrados  at  the  base  and  the  vertical  through 
the  centre  of  gravity. 


X1T  PREFACE. 

have  called  it  the  modulus  of  stability;  conceiving  thia 
measure  of  the  stability  to  be  of  more  obvious  and  easier 
application  than  the  coefficient  of  stability  used  by  the 
French  writers. 

That  structure  in  respect  to  every  independent  element 
of  which  the  modulus  of  stability  is  the  same,  is  evidently 
the  structure  of  the  greatest  stability  having  a  given  quantity 
of  material  employed  in  its  construction ;  or  of  the  greatest 
economy  of  material  having  a  given  stability. 

The  application  of  these  principles  of  construction  to  the 
theory  of  piers,  walls  supported  by  counterforts  and  shores, 
buttresses,  walls  supporting  the  thrust  of  roofs,  and  the 
weights  of  the  floors  of  dwellings,  and  Gothic  structures, 
has  suggested  to  me  a  class  of  problems  never,  I  believe, 
before  treated  mathematically. 

I  have  applied  the  well  known  principle  of  Coulomb  to 
the  determination  of  the  pressure  of  earth  upon  revetement 
walls,  and  a  modification  of  that  principle,  suggested  by  M. 
Poncelet,  to  the  determination  of  the  resistance  opposed  to 
the  overthrow  of  a  wall  backed  by  earth.  This  determina- 
tion has  an  obvious  application  to  the  theory  of  foundations. 

In  the  application  of  the  principle  of  Coulomb  I  have 
availed  myself,  with  great  advantage,  of  the  properties  of 
the  limiting  angle  of  resistance.  All  my  results  have  thus 
received  a  new  and  a  simplified  form. 

The  theory  of  the  arch  I  have  discussed  upon  principles 
first  laid  down  in  my  memoir  on  "  the  Theory  of  the  Stability 
of  a  System  of  Bodies  in  Contact,"  before  referred  to,  and 
subsequently  in  a  memoir  printed  in  the  "Treatise  on 
Bridges"  by  Professor  Hosking  and  Mr.  Hann.*  They 
differ  essentially  from  those  on  which  the  theory  of  Coulomb 
is  founded  ;f  when,  nevertheless,  applied  to  the  case  treated 

*  I  have  made  extensive  use  of  the  memoir  above  referred  to  in  the  following 
work,  by  the  obliging  permission  of  the  publisher,  Mr.  Weale. 

f  The  theory  of  Coulomb  was  unknown  to  me  at  the  time  of  the  publication 
of  my  memoirs  printed  in  the  Camb.  Phil.  Trans.  For  a  comparison  of  the 
two  methods  see  Mr.  Hann's  treatise. 


PKEFACE.  XT 

by  the  French  mathematicians,  they  lead,  to  identical  results, 
I  have  inserted  at  the  conclusion  of  my  work  the  tables  of 
the  thrust  of  circular  arches,  calculated  by  M.  Garidel  from 
formulae  founded  on  the  theory  of  Coulomb. 

The  fifth  part  of  the  work  treats  of  the  "strength  of 
materials,"  and  applies  a  new  method  to  the  determination 
of  the  deflexion  of  a  beam  under  given  pressures. 

In  the  case  of  a  beam  loaded  uniformly  over  its  whole 
length,  and  supported  at  four  different  points,  I  have  deter^ 
mined  the  several  pressures  upon  the  points  of  support  by  a 
method  applied  by  M.  Navier  to  a  similar  determination  in 
respect  to  a  beam  loaded  at  given  points.* 

In  treating  of  rupture  by  elongation  I  have  been  led  to  a 
discussion  of  the  theory  of  the  suspension  bridge.  This 
question,  so  complicated  when  reference  is  had  to  the  weight 
of  the  roadway  and  the  weights  of  the  suspending  rods,  and: 
when  the  suspending  chains  are  assumed  to  tte  of  uniform 
thickness,  becomes  comparatively  easy  when  the  section  of 
the  chain  is  assumed  so  to  vary  its  dimensions  as  to  be  every 
where  of  the  same  strength.  A  suspension  bridge  thus 
constructed  is  obviously  that  which,  being  of  a  given 
strength,  can  be  constructed  with  the  least  quantity  of 
materials ;  or,  which  is  of  the  greatest  strength  having  a 
given  quantity  of  materials  used  in  its  construction.! 

The  theory  of  rupture  by  transverse  strain  has  suggested 
a  new  class  of  problems,  having  reference  to  the  forms  of 
girders  having  wide  flanges  connected  by  slender  ribs  or  by 
open  frame  work  :  the  consideration  of  their  strongest  forms 
leads  to  results  of  practical  importance. 

In  discussing  the  conditions  of  the  strength  of  breast- 
summers,  my  attention  has  been  directed  to  the  best  positions 
of  the  columns  destined  to  support  them,  and  to  a  comparison 


*  As  in  fig.  p.  487.  of  the  following  work. 

f  That  particular  case  of  this  problem,  in  which  the  weights  of  the  suspending 
rods  are  neglected,  has  been  treated  by  Mr.  Hodgkinson  in  the  fourth  vol.  of 
Manchester  Transactions,  with  his  usual  ability.  He  has  not,  however,  suc- 
ceeded in  effecting  its  complete  solution. 


XVI  PREFACE. 

of  the  strength  of  a  beam  carrying  a  uniform  load  and  sup- 
ported freely  at  its  extremities,  with  that  of  a  beam  similarly 
loaded  but  having  its  extremities  firmly  imbedded  in 
masonry. 

In  treating  of  the  strength  of  columns  I  have  gladly 
replaced  the  mathematical  speculations  upon  this  subject, 
which  are  so  obviously  founded  upon  false  data,  by  the 
invaluable  experimental  results  of  Mr.  E.  Hodgkinson, 
detailed  in  his  well  known  paper  in  the  Philosophical 
Transactions  for  1840. 

The  sixth  and  last  part  of  my  work  treats  on  "  impact ;" 
and  the  Appendix  includes,  together  with  tables  of  the 
mechanical  properties  of  the  materials  of  construction,  the 
angles  of  rupture  and  the  thrusts  of  arches,  and  complete 
elliptic  functions,  a  demonstration  of  the  admirable  theorem 
of  M.  Poncelet  for  determining  an  approximate  value  of  the 
square  root  of  the  sum  or  difference  of  two  squares. 

In  respect  to  the  following  articles  of  my  work  I  have  tc 
acknowledge  my  obligations  to  the  work  of  M.  Poncelet, 
entitled  Mecanique  Industrielle.  The  mode  of  demonstration 
is  in  some,  perhaps,  so  far  varied  as  that  their  origin  might 
with  difficulty  be  traced ;  the  principle,  however,  of  each 
demonstration — all  that  constitutes  its  novelty  or  its  value — 
belongs  to  that  distinguished  author. 

30,*  38,  40,  45,  46,  47,  52,  58,  62,  75,  108,f  123,  202, 
267,t  268,  269,  270,  349,  354,  365.§ 

*  The  enunciation  only  of  this  theorem  is  given  in  the  Mec.  Ind.,  2me  partie, 
Art.  38. 

f  Some  important  elements  of  the  demonstration  of  this  theorem  are  taken 
from  the  Mec.  Ind.,  Art.  79.  2me  partie.  The  principle  of  the  demonstration 
is  not,  however,  the  same  as  in  that  work. 

\  In  this  and  the  three  following  articles  I  have  developed  the  theory  of  the 
9y-wheel,  under  a  different  form  from  that  adopted  by  M.  Poncelet  (Mec.  Ind., 
Art.  56.  3me  partie).  The  principle  of  the  whole  calculation  is,  however, 
taken  from  his  work.  It  probably  constitutes  one  of  the  most  valuable  of  hia 
contributions  to  practical  science. 

§  The  idea  of  determining  the  work  necessary  to  produce  a  given  deflection 
of  a  beam  from  that  expended  the  compression  and  the  elongation  of  its  com- 
ponent fibres  was  suggested  by  an  observation  in  the  Mec.  Ind.,  Art.  75.  3me 
partie. 


CONTENTS. 


STATICS. 

1*1 

The  Parallelogram  of  Pressures         ........  g 

The  Principle  of  the  Equality  of  Moments        ......  6 

The  Polygon  of  Pressures          .........  10 

The  Parallelopipedon  of  Pressures    ...        .....  14 

Of  Parallel  Pressures         ..........  16 

The  Centre  of  Gravity       ..........  20 

The  Properties  of  Guldinus       .........  3$ 


PART    II. 

DYNAMICS. 

Motion      ..............  47 

Velocity     .............  48 

WORK        .............  48 

Work  of  Pressures  applied  in  different  Directions  to  a  Body  moveable 

about  a  fixed  Axis          ..........  57 

Accumulation  of  Work      ..........  58 

Angular  Velocity       ...........  65 

The  Moment  of  Inertia     ..........  70 

THE  ACCELERATION  OF  MOTION  BY  GIVEN  MOVING  FORCES       .        .        .79 

The  Descent  of  a  Body  upon  a  Curve        .......  83 

The  Simple  Pendulum       ..........  85 

Impulsive  Force         ...........  86 

The  Parallelogram  of  Motion     .........  86 

The  Polygon  of  Motion      ..........  88 

The  Principle  of  D'Alembert     .........  89 

Motion  of  Translation        ..........  90 

Motion  of  Rotation  about  a  fixed  Axis       .......  91 

The  Centre  of  Percussion  .......        .        .96 

The  Centre  of  Oscillation  ..........  96 

Projectiles          ............  99 

Centrifugal  Force       ...........  106 

ft 


XV111  CONTENTS. 

Page 

The  Principle  of  virtual  Velocities 112 

The  Principle  of  Vis  Viva 115 

Dynamical  Stability * 121 

FRICTION 124 

Summary  of  the  Laws  of  Friction 130 

The  limiting  Angle  of  Resistance 131 

The  Cone  of  Resistance 133 

The  two  States  bordering  upon  Motion 133 

THE  RIGIDITY  OF  CORDS    .        .        .        .  « 142 


PART    III. 

THE   THEORY   OP   MACHINES. 

The  Transmission  of  Work  by  Machines    .                 146 

The  Modulus  of  a  Machine  moving  with  a  uniform  or  periodical  Motion    .  148 
The  Modulus  of  a  Machine  moving  with  an  accelerated  or  a  retarded 

Motion 150 

The  Velocity  of  a  Machine  moving  with  a  variable  Motion        .         .         .  151 
To  determine  the  Co-efficients  of  the  Modulus  of  a  Machine      .         .         .153 
General  Condition  of  the  State  bordering  upon  Motion  in  a  Body  acted 
upon  by  Pressures  in  the  same  Plane,  and  moveable  about  a  cylindrical 

Axis 154 

The  Wheel  and  Axle 155 

The  Pulley 160 

System  of  one  fixed  and  one  moveable  Pulley  .         .         .        .        .        .161 

A  System  of  one  fixed  and  any  Number  of  moveable  Pulleys   .        .         .163 

A  Tackle  of  any  Number  of  Sheaves 166 

The  Modulus  of  a  compound  Machine 169 

The  Capstan 194 

The  Chinese  Capstan 199 

The  Horse  Capstan,  or  the  Whim  Gin 202 

The  Friction  of  Cords 207 

The  Friction  Break 213 

The  Band 215 

The  modulus  of  the  Band 217 

The  Teeth  of  Wheels 227 

Involute  Teeth 234 

Epicycloidal  and  Hypocycloidal  Teeth 236 

To  set  out  the  Teeth  of  Wheels 239 

A  Train  of  Wheels 241 

The  Strength  of  Teeth 243 

To  describe  Epicycloidal  Teeth 245 

To  describe  involute  Teeth 251 

The  Teeth  of  a  Rack  and  Pinion        .                                                               .  253 


CONTENTS.  XLX 

r» 

Page 
The  Teefh  of  a  Wheel  working  with  a  Lantern  or  Trundle         .         .         .  25r< 

The  driving  and  working  Pressures  on  Spur  Wheels 259 

The  Modulus  of  a  System  of  two  Spur  Wheels  .         .         .         .        .        .  268 

The  Modulus  of  a  Rack  and  Pinion 282 

Conical  or  Bevil  Wheels 284 

The  Modulus  of  a  System  of  two  Bevil  Wheels 288 

The  Modulus  of  a  Train  of  Wheels 301 

The  Train  of  least  Resistance 310 

The  Inclined  Plane 312 

The  Wedge  driven  by  Pressure         .  321 

The  Wedge  driven  by  Impact 823 

The  mean  Pressure  of  Impact 325 

The  Screw 326 

Applications  of  the  Screw 329 

The  Differential  Screw 331 

Hunter's  Screw 332 

The  Theory  of  the  Screw  with  a  Square  Thread  in  reference  to  the  vari- 
able Inclination  of  the  Thread  at  different  Distances  from  the  Axis       .  333 

The  Beam  of  the  Steam  Engine 337 

The  Crank 341 

The  Dead  Points  in  the  Crank 845 

The  Double  Crank 346 

The  Crank  Guide 351 

The  Fly-wheel 353 

The  Friction  of  the  Fly-wheel 362 

The  Modulus  of  the  Crank  and  Fly-wheel 363 

The  Governor   . 364 

The  Carriage-wheel 368 

On  the  State  of  the  accelerated  or  retarded  Motion  of  a  Machine     .        .373 

PART    IV. 

THE    THEORY    OF    THE    STABILITY    OF    STRUCTURES. 

General  Conditions  of  the  Stability  of  a  Structure  of  Uncemented  Stones  87 7 

The  Line  of  Resistance      .  • 371 

The  Line  of  Pressure 37;) 

The  Stability  of  a  Solid  Body 38  > 

The  Stability  of  a  Structure 381 

The  Wall  or  Pier        .         .         . 382 

The  Line  of  Resistance  in  a  Pier 383 

The  Stability  of  a  Wall  supported  by  Shores 387 

The  Gothic  Buttress 396 

The  Stability  of  Walls  sustaining  Roofs      .......  397 

The  Plate  Band 402 

The  sloping  Buttress 40J 


XX  CONTENTS. 

Page 
The  Stability  of  a  Wall  sustaining  the  Pressure  of  a  Fluid        .        .        .408 

Earth  Works 412 

Revetement  Walls 416 

The  Arch 429 

The  Angle  of  Rupture 437 

The  Line  of  Resistance  in  a  circular  arch  whose  Voussoirs  are  equal,  and 
whose  Load  is  distributed  over  different  Points  of  its  Extrados      .         .  440 

A  segmental  Arch  whose  Extrados  is  horizontal 441 

A  Gothic  Arch,  the  Extrados  of  each  Semi- Arch  being  a  straight  Line 
inclined  at  any  given  Angle  to  the  Horizon,  and  the  Material  of  the 

Loading  different  from  that  of  the  Arch 442 

A  circular  Arch  having  equal  Voussoirs  and  sustaining  the  Pressure  of 

Water 444 

The  Equilibrium  of  an  Arch,  the  Contact  of  whose  Voussoirs  is  geometri- 
cally accurate 446 

Applications  of  the  Theory  of  the  Arch 448 

Tables  of  the  Thrust  of  Arches         ....  .454 


PART    V. 

THE   STRENGTH    OP   MATERIALS. 

Elasticity 458 

Elongation $  459 

The  Moduli  of  Resilience  and  Fragility 452 

Deflection .  467 

The  Deflexion  of  Beams  loaded  uniformly         ....  .481 

The  Deflexion  of  Breast  Summers .  486 

•Rupture 502 

Tenacity • 502 

The  Suspension  Bridge 505 

The  Catenary 50g 

The  Suspension  Bridge  of  greatest  Strength 510 

-  Rupture  by  Compression   .         .         .      * 618 

The  Section  of  Rupture  in  a  Beam     .....  520 

General  Conditions  of  the  Rupture  of  a  Beam 521 

The  Beam  of  greatest  Strength t  527 

The  Strength  of  Breast  Summers <  540 

The  best  Positions  of  their  Points  of  Support   ....  542 

Formulas  representing  the  absolute  Strength  of  a  Cylindrical  Column  to 

sustain  a  Pressure  in  the  Direction  of  its  Length 545 

Torsion 


CONTENTS. 


XXI 


PART    VI. 

IMPACT. 

Page 
The  Impact  of  two  Bodies  whose  centres  of  Gravity  move  in  the  same 

right  Line 553 

Greatest  Compression  of  the  Surface  of  the  Bodies   ...  .  555 

Velocity  of  two  elastic  Bodies  after  Impact        ...  .  556 

The  Pile  Driver 534 

ADDITIONS  BY  THE  AMERICAN  EDITOR        ....  .671 


APPENDIX. 

Note  A 631 

Note  B. — Poncelet's  Theorems 632 

Note  C. — On  the  Rolling  of  Ships 637 

Note  D 653 

Note  E. — On  the  Rolling  Motion  of  a  Cylinder 655 

Note  F. — On  the  Descent  upon  an  Inclined  Plane  of  a  Body  subject  to 
Variations  of  Temperature,  and  on  the  Motion  of  Glaciers     .         .         .  675 

Note  G. — The  best  Dimensions  of  a  Buttress 683 

Note  H. — Dimensions  of  the  Teeth  of  Wheels 684 

Note  I. — Experiments  of  M.  Morin  on  the  Traction  of  Carriages       .         .  685 

N"ote  K. — On  the  Strength  of  Columns  , 686 

Table  I. — The  Numerical  Values  of  complete  Elliptic  Functions  of  the 
first  and  second  Orders  for  Values  of  the  Modulus  Jc  corresponding  to 

each  Degree  of  the  Angle  $in-lk 687 

Table  II. — Showing  the  Angle  of  Rupture  •*•  of  an  Arch  whose  Loading 
is  of  the  same  Material  with  its  Voussoirs,  and   whose   Extrados  is 

inclined  at  a  given  Angle  to  the  Horizon 688 

Table  III. — Showing  the  Horizontal  Thrust  of  an  Arch,  the  Radius  of 
whose  Intrados  is  Unity,  and  the  Weight  of  each  Cubic  Foot  of  its 

Material  and  that  of  its  Loading,  Unity 691 

Table  IV. — Mechanical  Properties  of  the  Materials  of  Construction  .         .  694 
Table  V.— Useful  Numbers  .  698 


THE 


MECHANICAL  PRINCIPLES 


or 


CIVIL    ENGINEERING. 


PA.RT    I. 

STATICS, 

1.  FORCE  is  that  which,  tends  to  cause  or  to  destroy 
motion,  or  which  actually  causes  or  destroys-  it. 

The  direction  of  a  force  is  that  straight  line  in  which  it 
tends  to  cause  motion  in  the  point  to  which  it  is  applied,  or 
in  which  it  tends  to  destroy  the  motion  in  it.* 

When  more  forces  than  one  are  applied  to  a  body,  and 
their  respective  tendencies  to  communicate  motion  to  it 
counteract  one  another,  so  that  the  body  remains  at  rest, 
these  forces  are  said  to  be  in  EQUILIBRIUM,  and  are  called 
PRESSURES. 

It  is  found  by  experiment  f  that  the  effect  of  a  pressure, 
when  applied  to  a  solid  body,  is  the  same  at  whatever  point 
in  the  line  of  its  direction  it  is  applied ;  so  that  the  condi- 
tions of  the  equilibrium  of  that  pressure,  in  respect  to  other 
pressures  applied  to  the  same  body,  are  not  altered,  if,  with 
out  altering  the  direction  of  the  pressure,  we  remove  its. 
point  of  application,  provided  only  the  point  to  which  we 
remove  it  be  in  the  straight  line  in  the  direction  of  which  it 
acts. 

The  science  of  STATICS  is  that  which  treats  of  the  equili- 
brium of  pressures.  When  two  pressures  only  are  applied  to 

*  Note  (a)  Ed.  Appendix.  f  Note  (6)  Ed.  Appendix. 


THE    UNIT    OF    PKESSURE. 


a  body,  and  hold  it  at  rest,  it  is  found  by  experiment  that 
these  pressures  act  in  opposite  directions,  and  have  their 
directions  always  in  the  same  straight  line.  Two  such  pres- 
sures are  said  to  be  equal. 

If,  instead  of  applying  two  pressures  which  are  thus  equal 
in  opposite  directions,  we  apply  them  both  in  the  same 
direction,  the  single  pressure  which  must  be  applied  in  a 
direction  opposite  to  the  two  to  sustain  them,  is  said  to  be 
double  of  either  of  them.  If  we  take  a  third  pressure,  equal 
to  either  of  the  two  first,  and  apply  the  three  in  the  same 
direction,  the  single  pressure,  which  must  be  applied  in  a 
direction  opposite  to  the  three  to  sustain  them,  is  said  to  be 
triple  of  either  of  them :  and  so  of  any  number  of  pressures. 
Thus,  fixing  upon  any  one  pressure,  and  ascertaining  how 
many  pressures  equal  to  this  are  necessary,  when  applied  in 
an  opposite  direction,  to  sustain  any  other  greater  pressure, 
we  arrive  at  a  true  conception  of  the  amount  of  that  greater 
pressure  in  terms  of  the  first. 

That  single  pressure,  in  terma  of  which  the  amount  of  any 
other  greater  pressure  is  thus  ascertained,  is  called  an  UNIT 
of  pressure. 

Pressures,  the  amount  of  which  are  determined  in  terms 
of  some  known  unit  of  pressure,  are  said  to  be  measured. 

Different  pressures,  the  amounts  of  which  can  be  deter- 
mined in  terms  of  the  same  unit,  are  said  to  be  commensur- 
able. 

The  units  of  pressure  which  it  is  found  most  convenient  to 
use,  are  the  weights  of  certain  portions  of  matter,  or  the 
pressures  with  which  they  tend  towards  the  centre  of  the 
earth.  The  units  of  pressure  are  different  in  different  coun- 
tries. With  us,  the  unit  of  pressure  from  which  all  the  rest 
are  derived  is  the  weight  of  22-S15*  cubic  inches  of  distilled 
water.  This  wreiglit  is  one  pound  troy ;  being  divided  into 
5760  equal  parts,  the  weight  of  each  is  a  grain  troy,  and 
TOGO  such  grains  constitute  the  pound  avoirdupois. 

If  straight  lines  be  taken  in  the  directions  of  any  number 
of  pressures,  and  have  their  lengths  proportional  to  the 
numbers  of  units  in  those  pressures  respectively,  then  these 
lines  having  to  one  another  the  same  proportion  in  length 
that  the  pressures  have  in  magnitude,  and  being  moreover 
draw^n  in  the  directions  in  which  those  pressures  respectively 
act,  are  said  to  represent  them  in  magnitude  and  direction. 

*  This  standard  was  fixed  by  Act  of  Parliament,  in  1824.  The  temperature 
of  the  water  is  supposed  to  be  62°  Fahrenheit,  the  weight  to  be  taken  in  air, 
and  the  barometer  to  stand  at  30  inches. 


THE   PARALLELOGRAM   OF   PRESSURES.  3 

A  system  of  pressures  being  in  equilibrium,  let  any  num- 
ber of  them  be  imagined  to  be  taken  away  and  replaced  by 
a  single  pressure,  and  let  this  single  pressure  be  such  that 
the  equilibrium  which  before  existed  may  remain,  then  this 
single  pressure,  producing  the  same  effect  in  respect  to  the 
equilibrium  that  the  pressures  which  it  replaces  produced,  is 
said  to  be  the  RESULTANT. 

The  pressures  which  it  replaces  are  said  to  be  the  COMPO- 
NENTS of  this  single  pressure ;  and  the  act  of  replacing  them 
by  such  a  single  pressure,  is  called  the  COMPOSITION  of 
pressures. 

If,  a  single  pressure  being  removed  from  a  system  in  equi- 
librium, it  be  replaced  by  any  number  of  other  pressures, 
such,  that  whatever  effect  was  produced  by  that  which  they 
replace  singly,  the  same  effect  (in  respect  to  the  conditions  of 
the  equilibrium)  may  be  produced  by  those  pressures  con- 
jointly, then  is  that  single  pressure  said  to  have  been  RE- 
SOLVED into  these,  and  the  act  of  making  this  substitution 
of  two  or  more  pressures  for  one,  is  called  the  RESOLUTION 
of  pressures. 

THE  PARALLELOGRAM  OF  PRESSURES. 

2.  The  resultant  of  any  two  pressures  applied  to  a  point, 
is  represented  in  direction  by  the  diagonal  of  a  paral- 
lelogram, whose  adjacent  sides  represent  those  pressures  in 
magnitude  and  direction* 

(Duchayla's  Method.f) 

To  the  demonstration  of  this* proposition,  after  the  excel- 
lent method  of  Duchayla,  it  is  necessary  in  the  first  place 
to  show,  that  if  there  be  any  two  pressures  P2  and  P3  whose 
directions  are  in  the  same  straight  line,  and  a  third  pressure 
Px  in  any  other  direction,  and  if  the  proposition  be  true  in 
respect  to  Pj  and  P2,  and  also  in  respect  to  P1  and  P3,  then 
it  will  be  true  in  respect  to  Pj  and  P2-f-P3« 

Let  P15  P2,  and  P3,  form  part  of  any  system  of  pressures  in 

?    ?     equilibrium,  and  let  them  be  applied  to  the  point 

^;*C%;Tr\     A;  take  AB  and  AC  to  represent,  in  magnitude 

\   x\;V^v,   and  direction,  the  pressures  Yl  and  PQ,  and  CD 

>-£."»*  foQ  pressTire  P3?  and  complete  the  parallelograms 

CB  and  DF.     Suppose  the  proposition  to  be  true  with  regard 

*  This  proposition  constitutes  the  foundation  of  the  entire  science  of  Statics. 
f  Note  (c)  Ed.  App. 


4:  THE   PAKALLELOGRAM 

to  P,  and  P2,  tlie  resultant  of  'Pl  and  P2  will  then  be  in  the 
direction  of  the  diagonal  AF  of  the  parallelogram  BO,  whose 
adjacent  sides  AC  and  AB  represent  P,  and  P2  in  magnitude 
and  direction.  Let  P,  and  P?  be  replaced  by  this  resultant. 
It  matters  not  to  the  equilibrium  where  in  the  line  AF  it  is 
applied ;  let  it  then  be  applied  at  F.  But  thus  applied  at 
F  it  may,  without  affecting  the  conditions  of  the  equilibrium, 
be  in  its  turn  replaced  by  (or  resolved  into)  two  other  pressures 
acting  in  OF  and  BF,  and  these  will  manifestly  be  equal  to 
P,  and  P2,  of  which  P,  may  be  transferred  without  altering 
the  conditions  to  0,  and  P2  to  E.  Let  this  be  done,  and  let 
P3  be  transferred  from  A  to  C,  we  shall  then  have  Pj  and 
!P3  acting  in  the  directions  CF  and  CD  at  C  and  P2,  in  the 
direction  FE  at  E,  and  the  conditions  of  the  equilibrium  will 
not  have  been  affected  by  the  transfer  of  them  to  these 
points.  .Now  suppose  that  the  proposition  is  also  true  in 
respect  to  Px  and  P3  as  well  as  Px  and  P2.  Then  since  CF 
and  CD  represent  ttl  and  P3  in  magnitude  and  direction, 
therefore  their  resultant  is  in  the  direction  of  the  diagonal 
CE.  Let  them  be  replaced  by  this  resultant,  and  let  it  be 
transferred  to  E,  and  let  it  then  be  resolved  into  two  other 
pressures  acting  in  the  directions  DE  and  FE ;  these  will 
evidently  be  Pa  and  P8.  We  have  now  then  transferred  all 
the  three  pressures  P1?  P2,  P3,  from  A  to  E,  an.d  they  act  at  E 
in  directions  parallel  to  the  directions  in  which  they  acted  at 
A,  and  this  has  been  done  without  affecting  the  conditions  of 
the  equilibrium ;  or,  in  other  words,  it  has  been  shown  that 
the  pressures  P1?  P2,  P3,  produce  the  same  effect  as  it  re- 
spects the  conditions  of  the  equilibrium,  whether  they  be 
applied  at  A  or  E.  The  residtant  of  P1?  T2,  P3,  must  there- 
fore produce  the  same  effect  as  it  regards  the  conditions  of 
the  equilibrium,  whether  it  be  applied  at  A  or  E.  But  in 
order  that  this  resultant  may  thus  produce  the  same  effect 
when  acting  at  A  or  E,  it  must  act  in  the  straight  line  AE, 
because  a  pressure  produces  the  same  effect  when  applied  at 
two  different  points  only  when  both  those  points  are  in  the 
line  of  its  direction.  On  the  supposition  made,  therefore, 
the  resultant  of  P,,  P2,  and  P3,  or  of  P,  and  P2  +  P3 
acts  in  the  direction  of  the  diagonal  AE  of  the  parallel- 
ogram BD,  whose  adjacent  sides  AD  and  AB  represent 
Pa  +  P3  and  P>  in  magnitude  and  direction ;  and  it  has  been 
shown,  that  if  the  proposition  be  true  in  respect  to  Pt  and 
P2,  and  also  in  respect  to  P,  and  P3,  then  it  is  true  in  respect 
to  Pj  and  Pa  +  P3.  Now  this  being  the  case  for  all  values 
°f  P«  Pa,  P35  it  is  the  case  when  P1?  P2,  and  P8,  are  equal 


OF    PRESSURES*  5 

to  one  another.  But  if  Px  be  equal  to  Pa  their  resultant 
will  manifestly  have  its  direction  as  much  towards  one  of 
these  pressures  as  the  other ;  that  is,  it  will  have  its  direc- 
tion midway  between  them,  and  it  will  bisect  the  angle  BAG : 
but  the  diagonal  AF  in.  this  case  also  bisects  the  angle  BAG, 
since  P,  being  equal  to  P2,  AC  is  equal  to  AB ;  so  that  in 
this  particular  case  the  direction  of  the  resultant  is  the 
direction  of  the  diagonal,  and  the  proposition  is  true,  and 
similarly  it  is  true  of  Pt  and  P3,  since  these  pressures  are 
equal.  Since  then  it  is  true  of  Pl  and  P2  when  they  are 
equal,  and  also  of  Px  and  P3,  therefore  it  is  true  in  this  case 
of  P,  and  P2  +  Ps,  that  is  of  P1  and  2  Pr  And  since  it  is 
true  of  Pj  and  P2,  and  also  of  P1  and  2  Pn  therefore  it  is  true 
of  _P1  and  P2  +  2  P15  that  is  of  P,  and  3  P, ;  and  so  of  P,  and 
m  P15  if  m  be  any  w^hole  number ;  and  similarly  since  it  is 
true  of  m  Px  and  P1?  therefore  it  is  true  of  m  Pa  and  2  P,,  &c., 
and  of  m  Pt  and  n  P1  where  n  is  any  whole  number.  There- 
fore the  proposition  is  true  of  any  two  pressures  raPx  and 
n  P1  which  are  commensurable. 

It  is  moreover  true  when  the  pressures  are  in- 
j,^.........^  cot)imensuraiie^     yor  iet  AC  anci  AB  represent 

|:;V:~\y/r:-::i?  anv  two  such  pressures  P!  and  P2  in  magnitude 
and  direction,  and  complete  the  parallelogram 
ABDC,  then  will  the  direction  of  the  resultant  of  P,  and 
P2  be  in  AD ;  for  if  not,  let  its  direction  be  AE,  and  draw 
EG  parallel  to  CD.  Divide  AB  into  equal  parts,  each  less 
than  GO,  and  set  oif  on  AC  parts  equal  to  those  from  A 
towards  C.  One  of  the  divisions  of  these  will  manifestly 
fall  in  GC.  Let  it  be  H,  and  complete  the  parallelogram 
AHFB.  Then  the  pressure  P2  being  conceived  to  be 
divided  into  as  many  equal  units  of  pressure  as  there  are 
equal  parts  in  the  line  AB,  AH  may  be  taken  to  represent  a 
pressure  P3  containing  as  many  ot  these  units  of  pressure 
as  there  are  equal  parts  in  AH,  and  these  pressures  P2  and 
P3  will  be  commensurable,  being  measured  in  terms  of  the 
same  unit.  Their  resultant  is  therefore  in  the  direction  AF, 
and  this  resultant  of  P3  and  P2  has  its  direction  nearer  to 
AC  than  the  resultant  AE  of  P,  and  P2  has ;  which  is 
absurd,  since  Pa  is  greater  than  P3. 

Therefore  AE  is  not  in  the  direction  of  the  resultant  ot 
P  and  Pa ;  and  in  the  same  manner  it  may  be  shown  that  no 
other  than  AD  is  in  that  direction.  Therefore,  &c. 


THE   PRINCIPLES    OF   THE 


3.  The  resultant  of  two  pressures  applied  in  any  directions 
to  a  point,  is  represented  in  magnitude  as  well  as  in  direc- 
twnoy  the  diagonal  of  the  parallelogram  whose  adjacent 
sides  represent  those  pressures  in  magnitude  and  in  direc- 
tion. 

Let  BA  and  CA  represent,  in  magnitude  and 
%.,  direction,  any  two  pressures  applied  to  the  point 

"  A.     Complete  the  parallelogram  BC.     Then  by 

the  last  proposition  AD  will  represent  the  result- 
ant  of  these  pressures  in  direction.  It  will  also 
represent  it  in  magnitude ;  for,  produce  DA  to  G,  and  con- 
ceive a  pressure  to  be  applied  in  GA  equal  to  the  resultant 
of  BA  and  CA,  and  opposite  to  it,  and  let  this  pressure  be 
represented  in  magnitude  by  the  line  GA.  Then  will  the 
pressures  represented  by  the  lines  BA,  CA,  and  GA,  mani- 
festly be  pressures  in  equilibrium.  Complete  the  parallelo- 
gram BGr  then  is  the  resultant  of  GA  and  BA  in  the 
direction  FA;  also  since  GA  and  BA  are  in  equilibrium 
wTith  CA,  therefore  this  resultant  is  in  equilibrium,  with  CA, 
but  when  two  pressures  are  in  equilibrium,  their  directions 
are  in  the  same  straight  line ;  therefore  FAC  is  a  straight 
line.  But  AC  is  parallel  to  BD,  therefore  FA  is  parallel  to 
BD,  and  FB  is,  by  construction,  parallel  to  GD,  therefore 
AFBD  is  a  parallelogram,  and  AD  is  equal  to  FB  and 
therefore  to  AG.  But  AG  represents  the  resultant  of  CA 
and  BA  in  magnitude,  AD  therefore  represents  it  in  magni- 
tude. Therefore,  &c* 


THE  PRINCIPLE  OF  THE  EQUALITY  OF  MOMENTS. 

4.  DEFINITION.  If  any  number  of  pressures  act  in  the 
same  plane,  and  any  point  be  taken  in  that  plane,  and  per- 
pendiculars be  drawn  from  it  upon  the  directions  of  all  these 
pressures,  produced  if  necessary,  and  if  the  number  of  units 
in  each  pressure  be  then  multiplied  by  the  number  of  units 
in  the  corresponding  perpendicular,  then  this  product  is 
called  the  moment  of  that  pressure  about  the  point  from 
which  the  perpendiculars  are  drawn,  and  these  moments  are 
said  to  be  measured  from  that  point. 

*  Note  (d)  Ed.  App. 


EQUALITY   OF   MOMENTS. 


5.  If  three  pressures  be  in  equilibrium,  and  their  moments 
~be  taken  about  any  point  in  the  plane  in  which  they  act, 
then  the  sum  of  the  moments  of  those  two  pressures  which 
tend  to  turn  the  plane  in  one  direction  about  the  point 
from  which  the  moments  are  measured,  is  equal  to  the 
moment  of  that  pressure  which  tends  to  turn  it  in  the 
opposite  direction. 


.  .....  *,c  P15  Pa,  P3,   acting  in  the    directions 

PA.PA  p30,   be   any  three  pressures  in 
~D,....-^>|--iB  equilibrium.     Take  any  point  A  in  the  plane 

'*-"'  in  which  they  act,  and  measure  their  moments 

from  A,  then  will  the  sum  of  the  moments  of  P2  and  P8, 
which  tend  to  turn  the  plane  in  one  direction  about  A,  equal 
the  moment  of  P1?  which  tends  to  turn  it  in  the  opposite 
direction. 

Through  A  draw  DAB  parallel  to  OP15  and  produce  OP, 
to  meet  it  in  D.  Take  OD  to  represent  P3,  and  take  DB 
such  a  length  that  OD  may  have  the  same  proportion  to 
DB  that  P2  has  to  P,.  Complete  the  parallelogram  ODBC, 
then  will  OD  and  OC  represent  P2  and  P1  in  magnitude  and 
direction.  Therefore  OB  will  represent  P3  in  magnitude 
and  direction. 

Draw  AM,  AN,  AL,  perpendiculars  on  OC,  OD,  OB, 
and  join  AO,  AC.  Now  the  triangle  OBC  is  equal  to  the 
triangle  OAC,  since  these  triangles  are  upon  the  same  base 
and  between  the  same  parallels. 

Also,    A  ODA+  AOAB  =  AOBD  =  AOBC, 
.-.A  PDA  -f  AO  AB=  A  OAC, 


AN  +  P3x  AL=Px  AM. 


Now  Pj  x  AM,  P2  x  AN,  P3  x  AL,  are  the  moments  of  P,, 
P2,  P3,  about  A  (Art.  4.) 

.•.mtP9  +  mtP>  =  mtP1     ......     (1). 

Therefore,  &c.  &c. 

6.  If  E  be  the  resultant  of  P2  and  P^then  since  E  is 
equal  to  P1  and  acts  in  the  same  straight  line,  rr^E  =  mtPj, 


The  sum  of  the  moments  therefore,  about  any  point,  of 
two  pressures,  Pa  and  P3  in  the  same  plane,  which  tend  to 


O  THE   PRINCIPLE   OF   THE 

turn  it  in  the  same  direction  about  that  point,  is  equal  to 
the  moment  of  their  resultant  about  that  point. 

If  they  had  tended  to  turn  it  in  opposite  directions,  then 
the  difference  of  their  moments  would  have  equalled  the 
moment  of  their  resultant.  For  let  R  be  the  resultant  of 
Pt  and  P3,  which  tend  to  turn  the  plane  in  opposite  direc- 
tions about  A,  &c.  Then  is  R  equal  to  P2,  and  in  the  same 
straight  line  with  it,  therefore  moment  R  is  equal  to 
moment  P2.  But  by  equation  (1)  m.tP1 — m'P,,  =  m^ ; 
.-.mT,— mtP3  =  mtR. 

Generally,  therefore,  m*  P,  4-  m*  P2  =  m1  R (2), 

the  moment,  therefore,  of  the  resultant  of  any  two  pressures 
in  the  same  plane  is  equal  to  the  sum  or  difference  of  the 
moments  of  its  components,  according  as  they  act  to  turn  the 
plane  in  the  same  direction  about  the  point  from  which  the 
moments  are  measured,  or  in  opposite  directions.* 

7.  If  any  number  of  pressures  in  the  same  plane  be  in  equi- 
librium* and  any  point  be  taken,  in  that  plane,  from 
which  their  moments  are  measured,  then  the  sum  of  the 
moments  of  those  pressures  which  tend  to  turn  the  plane 
in  one  direction  about  that  point  is  equal  to  the  sum  of  the 
moments  of  those  which  tend  to  turn  it  in  the  opposite 
direction. 

Let  P15  Pa,  P3 P«,  be  any  number  of  pressures  in 

the  same  plane  which  are  in  equi- 
librium, and  A  any  point  in  the 
plane  from  which  their  moments 
are  measured,  then  will  the  .sum  of 
the  moments  of  those  pressures 
which  tend  to  turn  the  plane  in  one  direction  about  A  equal 
the  sum  of  the  moments  of  those  which  tend  to  turn  it  in 
the  opposite  direction. 

Let  R,  be  the  resultant  of  P1  and  P2, 

R2 R,  and  P3, 

R3 Ra  and  P4, 

&c &c. 


Therefore,  by  the  last  proposition,  it  being  understood 
that  the  moments  of  those  of  the  pressures  r1?  P2,  which 
)  to  the  left  of  A 

*  Note  (c)  Ed.  App. 


tend  to  turn  the  plane  to  the  left  of  A,  are  to  be  taken  nega- 
tively, we  have 


EQUALITY    OF   MOMENTS. 

m*  K,  =  m4  Pt  +  m*  P2. 

m*  E2  =  m*  E!  +  m*  P,, 

m4  E3  =  m*  K2  +  m'  P4, 

&c.    =    &c.  &c. 

m*  En_    =  m* 


Adding  these  equations  together,  and  striking  out  the 
terms  common  to  both  sides,  we  have 


m*  P,  +  in*  P2  4-  m1  P3  -f   .....  .+  m*  Pn 

(3),  where  Rn_i  is  the  resultant  of  all  the  pressures  P1? 


But  these  pressures  are  in  equilibrium  ;  they  have,  there- 
fore, no  resultant. 

.-.Kn-i  =  0  .-.  m'En-!  =  0, 
.-.m4  P,  +  m*  P2  +  m1  P3,  +  .....  m*  Pn  =  0  .  .  .  .  (4). 

Now,  in  this  equation  the  moments  of  those  pressures  which 
tend  to  turn  the  system  to  the  left  hand  are  to  be  taken 
negatively.  Moreover,  the  sum  of  the  negative  terms  must 
equal  the  sum  of  the  positive  terms,  otherwise  the  whole 
sum  could  not  equal  zero.  It  follows,  therefore,  that  the 
sum  of  the  moments  of  those  pressures  which  tend  to  turn 
the  system  to  the  right  must  equal  the  sum  of  the  moments 
of  those  which  tend  to  turn  it  to  the  left.  Therefore,  &c.  &c. 


8.  If  any  number  of  pressures  acting  in  the  same  plane  fie  in 
equilibrium,  and  they  be  imagined  to  be  moved  parallel^  to 
their  existing  directions,  and  all  applied  to  the  samepoint^ 
so  as  all  to  act  upon  that  point  ^n  directions  parallel  to 
those  in  which  they  before  acted  upon  different  points,  then 
will  they  he  in  equilibrium  about  that  point. 

For  (see  the  preceding  figure)  the  pressure  E,  at  whatever 
point  in  its  direction  it  be  conceived  to  be  applied,  may  be 
resolved  at  that  point  into  two  pressures  parallel  and  equal 
to  Pj  and  P2 :  similarly,  E2  may  be  resolved,  at  any  point  in 
its  direction,  into  two  pressures  parallel  and  equal  to  Ej  and 
P3,  of  which  Ex  may  be  resolved  into  two,  parallel  and  equal 
to  P,  and  P2,  so  that  E2  may  be  resolved  at  any  point  of  its 
direction  into  three  pressures  parallel  and  equal  to  Pn  P2,  P8: 
and,  in  like  manner,  E3  may  be  resolved  into  two  pressures 
parallel  and  equal  to  E2  and  P4,  and  therefore  into  four  pres- 
sures parallel  and  equal  to  P1?  P2,  P3,  P4,  and  so  of  the  rest. 


10  THE   POLYGON 

Therefore  K«_i  may,  at  any  point  of  its  direction  be  resolved 

into  n  pressures  parallel  and  equal  to  Pn  P2,  P3, Pn ; 

if,  therefore,  n  such  pressures  were  applied  to  that  point, 
they  would  just  be  held  in  equilibrium  by  a  pressure  equal 
and  opposite  to  Rn_ i.  But  R»_i  =  0;  these  n  pressures 
would,  therefore,  be  in  equilibrium  with  one  another  if 
applied  to  this  point. 

Now  it  is  evident,  that  if,  being  thus  applied  to  this  point, 
they  would  be  in  equilibrium,  they  would  be  in  equilibrium 
if  similarly  applied  to  any  other  point.  Therefore,  &c. 


THE  POLYGON  OF  PRESSURES. 

9.  The  conditwns  of  the  equilibrium  of  any  number  of  pres- 
sures applied  to  a  point. 

Let  OP15  OP2,  OP3,  &c.,  represent  in  mag- 
nitude and  direction  pressures  Px,  P2,  &c., 
applied  to  the  same  point  O.  Complete  the 
parallelogram  OPj  AP2,  and  draw  its  diago- 
nal OA ;  then  will  OA  represent  in  magni- 
tude and  direction  the  resultant  of  Pj  and 
P2.  Complete  the  parallelogram  OABP3,  then  will  OB 
represent  in  magnitude  and  direction  the  resultant  of  OA 
and  P3 ;  but  OA  is  the  resultant  of  Px  and  P2,  therefore  OB 
is  the  resultant  of  P1?  P2,  P3 ;  similarly,  if  the  parallelogram 
OBCP4  be  completed,  its  diagonal  OC  represents  the  result- 
ant of  OB  and  P4,  that  is,  of  P,,  P2,  P3,  P4,  and  in  like 
manner  OD,  the  diagonal  of  the  parallelogram  OCDP5, 
represents  the  resultant  of  P,,  P2,  P3,  P4,  PB. 

ISTow  let  it  be  observed,  that  APX  is  equal  and  parallel  to 
OP2,  AB  to  OP3,  BO  to  OP4,  CD  to  OP.,  so  that  P,A,  AB, 
BC,  CD,  represent  P2,  P3,  P4,  P6,  respectively  in  magnitude, 
and  are  parallel  to  their  directions.  Moreover,  OPX  is  in  the 
direction  of  Pj  and  represents  it  in  magnitude,  so  that  the 
sides  OP,,  P,A,  AB,  BC,  CD,  of  the  polygon  OP1?  ABCDO 
represent  the  pressures  Pa,  P2,  P8,  P4,  P5,  respectively  in 
magnitude,  and  are  parallel  to  their  directions ;  whilst  the 
side  OD,  which  completes  that  polygon,  represents  the 
resultant  of  those  pressures  in  magnitude  and  direction. 

If,  therefore,  the  pressures  P0  P2,  P8,  P4,  P5,  be  in  equili- 
brium, so  that  they  have  no  resultant,  then  the  side  OD  of 
the  polygon  must  vanish,  and  the  point  D  coincide  with  O. 
Thus,  then,  if  any  number  of  pressures  be  applied  to  a  point 


OF   PRESSURES.  H 

and  lines  be  drawn  parallel  to  the  directions  of  those  pres- 
sures, and  representing  them  in  magnitude,  so  as  to  form 
sides  of  a  polygon  (care  being  taken  to  draw  each  line  from 
the  point  where  it  unites  with  the  preceding,  towards  the 
direction  in  which  the  corresponding  pressure  acts),  then  the 
line  thus  drawn  parallel  to  the  last  pressure,  and  representing 
it  in  magnitude,  will  pass  through  the  point  from  which  the 
polygon  commenced,  and  will  just  complete  it  if  the  pres- 
sures be  in  equilibrium ;  and  if  they  be  not  in  equilibrium, 
then  this  last  line  will  not  complete  the  polygon,  and  if  a 
line  be  drawn  completing  it,  that  line  will  represent  the 
resultant  of  all  the  pressures  in  magnitude  and  direction. 

This  principle  is  that  of  the  POLYGON  OF  PRESSURES  ;  it 
obtains  in  respect  to  pressures  applied  to  the  same  point, 
whether  they  be  in  the  same  plane  or  not. 

10.  If  any  number  of  pressures  in  the  same  plane  "be  in  equi- 
librium, and  each  be  resolved  in  directions  parallel  to  any 
two  rectangular  axes,  then  the  sum  of  all  those  resolved 
pressures,  whose  tendency  is  to  communicate  motion  in  one 
direction  along  either  axis,  is  equal  to  the  sum  of  those 
whose  tendency  is  in  the  opposite  direction. 

Let  the  polygon  of  pressures  be  formed  in  respect  to  any 
number  of  pressures,  Pn  PQ,  P3,  P4,  in  the  same  plane  and  in 
equilibrium  (Arts.  8,  9),  and  let  the  sides  of 
^"s  P°tyg°n  be  projected  on  any  straight  line 
Ax  in  the  same  plane.  ]STow  it  is  evident, 
that  the  sum  of  the  projections  of  those  sides 
&•  of  the  polygon  which  form  that  side  of  the 
figure  which  is  nearest  to  Ax,  is  equal  to  the  sum  of  the  pro- 
jections of  those  sides  which  form  the  opposite  side  of  the 
polygon :  moreover,  that  the  former  are  those  sides  of  the 
polygon  which  represent  pressures  tending  to  communicate 
motion  from  A  towards  x,  or  from  left  to  right  in  respect  to 
the  line  Ax ;  and  the  latter,  those  which  tend  to  communi- 
cate motion  in  the  opposite  direction.  Now  each  projection 
is  equal  to  the  corresponding  side  of  the  polygon,  multiplied 
by  the  cosine  of  its  inclination  to  Ax.  The  sum  of  all  those 
sides  of  the  polygon  which  represent  pressures  tending  to 
communicate  motion  from  A  towards  x,  multiplied  each  by 
the  cosine  of  its  inclination  to  Ax,  is  equal,  therefore,  to  the 
sum  of  all  the  sides  representing  pressures  whose  tendency 
is  in  the  opposite  direction,  each  being  similarly  multiplied 
by  the  cosine  of  its  inclination  to  Ax.  Now  the  sides  of  the 


12  THE   RESOLUTION 

polygon  represent  the  pressures  in  magnitude,  and  are 
inclined  at  the  same  angles  to  Ax.  Therefore,  each  pressure 
being  multiplied  by  the  cosine  of  its  inclination  to  Aa?,  the 
sum  of  all  these  products,  in  respect  to  those  which  tend  to 
communicate  motion  in  one  direction,  equals  the  sum  simi- 
larly taken  in  respect  to  those  which  tend  to  communicate 
motion  in  the  opposite  direction ;  or,  if  in  taking  this  sum  it 
be  understood  that  each  term  into  which  there  enters  a  pres- 
sure, whose  tendency  is  from  A  towards  a?,  is  to  be  taken 
positively,  whilst  each  into  which  there  enters  a  pressure 
which  tends  from  x  towards  A  is  to  be  taken  negatively, 
then  the  sum  of  all  these  terms  will  equal  zero ;  that  is, 
calling  the  inclinations  of  the  directions  of  P15  P2,  P3  .  .  .  P4 
to  Aa?,  «1?  «2,  «3  .  .  .  .  an  respectively, 

P,  cos.  «>  -f  P2cos.«2  +  P3cos.«3  + +  Pn  cos.  an  =0  ...  (5), 

in  which  expression  all  those  terms  are  to  be  taken  nega- 
tively which  include  pressures,  whose  tendency  is  from  x 
towards  A. 

This  proposition  being  true  in  respect  to  any  axis,  Aa?  is 
true  in  respect  to  another  axis,  to  which  the  inclinations  of 
the  directions  of  the  pressures  are  represented  by  /315  j32,  /33, 
Pn ,  so  that, 

P,  cos.  ft  +  P2  cos.  ft  +  .  .  .  .  +  P^cos.  pn  =0. 
Let  this  second  axis  be  at  right  angles  to  the  first : 

then  ft  =  —  —  ax  /.  cos.  ft  =  sin.  o1?  j3a=  — —  «2,  .%  cos.  ft 

=  sin.  aa,  &c.  =  &c. 

/.  P,  sin.  ax  +  P2  sin.  «2  +  ....+  Pwsin.  an  =  0  .  .  .  .  (6)  ; 

those  terms  in  this  equation,  involving  pressures  which  tend 
to  communicate  motion  in  one  direction,  in  respect  to  the 
axis  Ay  being  taken  with  the  positive  sign,  and  those  which 
tend  in  the  opposite  direction  with  the  negative  sign. 

If  the  pressures  P15  P2,  &c.  be  each  of  them  resolved 
into  two  others,  one  of  which  is  parallel  to  the  axis  Aa?,  and 
the  other  to  the  axis  Ay,  it  is  evident  that  the  pressures 
thus  resolved  parallel  to  Aa?,  will  be  represented  by  P,  cos.  al5 
P2  cos.  a,,  &c.,  and  those  resolved  parallel  to  Ay,  by 
P!  sin.  a,  P2  sin.  aa,  &c.  Thus  then  it  follows,  that  if 
any  system  of  pressures  in  equilibrium  be  thus  resolved 
parallel  to  two  rectangular  axes,  the  sum  of  those  resolved 
pressures,  whose  tendency  is  in  one  direction  along  either 


OF   PRESSURES.  13 

axis,  is  equal  to  the  sum  of  those  whose  tendency  is  in  the 
opposite  direction.* 

This  condition,  and  that  pf  the  equality  of  moments,  are 
necessary  to  the  equilibrium  of  any  number  of  pressures  in 
the  same  plane,  and  they  are  together  sufficient  to  that  equi- 
librium. 

11.  To  determine  the  resultant  of  any  number  of  pressures 
in  the  same  plane. 

If  the  pressures  Pj  Pa  ....  Pw  be  not  in 
equilibrium,  and  have  a  resultant,  then  one 
side  is  wanting  to  complete  the  polygon  of 
pressures,  and  that  side  represents  the  res- 
ultant of  all  the  pressures  in  magnitude, 
and  is  parallel  to  its  direction  (Art.  9). 
Moreover  it  is  evident,  that  in  this  case  the  sum  of  the  pro- 
jections on  Ax  (Art.  10)  of  those  lines  which  form  one 
side  of  the  polygon,  will  be  deficient  of  the  sum  of  those  of 
the  lines  which  form  the  other  side  of  the  polygon,  by  the 
projection  of  this  last  deficient  side  ;  and  therefore,  that  the 
sum  of  the  resolved  pressures  acting  in  one  direction  along 
the  line  A#,  will  be  less  than  the  sum  of  the  resolved  pres- 
sures in  the  opposite  direction,  by  the  resolved  part  of  the 
resultant  along  this  line.  Now  if  R  represent  this  resultant, 
and  6  its  inclination  to  AOJ,  then  R  cos.  0  is  the  resolved  part 
of  R  in  the  direction  of  A.X.  Therefore  the  signs  of  the  terms 
being  understood  as  before,  we  have 

R  cos.  0=Pa  cos.  OJ  +  PS  cos.  «„+....  +Pwcos.  an  .  .  (7). 
And  reasoning  similarly  in  respect  to  the  axis  Ay,  we  have 
R  sin.  0=P1  sin.  a.  +  P,  sin.  a2+  ....  +Pnsin.  an  .  .  .  (8). 


Squaring  these  equations  and  adding  them,  and  observing 
that  R2  sin.2  0-f  R2cos.2  0—  Ra  (sin.2  0+cos.a0)  =R2,  we  have 


R2=(2P  sin.  «)2  +  (2P  cos.  a)2  ........  (9), 

where  2P  sin.  a  is  taken  to  represent  the  sum  P,  sin.  ^  -r 
P2  sin.  aa  +  P3  sin.  as  +  &c.,  and  2p  cos.  a   to   represent   the 
sum  Px  cos.  o^+P,,  cos.  «2  +  P3  cos.  a3+  &c. 
Dividing  equation  (8)  by  equation  (7), 

r,     2P  sin.  a  /im 

tan.  0=-_—       -  .......  (10). 

SP  cos.  a 

Thus  then  by  equation  (9)  the  magnitude  of  the  resultant 
»  Note(/)Ed.  App. 


14:  TIIE    PARALLELOPIPEDON 

K  is  known,  and  by  equation  (10)  its  inclination  6  to  the  axis 
Ax  is  known.  In  order  completely  to  determine  it,  we  have 
yet  to  find  the  perpendicular  distance  at  which  it  acts  from 
the  given  point  A.  For  this  wre  must  have  recourse  to  the 
condition  of  the  equality  of  moments  (Art.  Y). 

If  the  sum  of  the  moments  of  those  of  the  pressures,  P1? 
Pa  .  .  .  .  P«,  ,  which  tend  to  turn  the  system  in  one  direc- 
tion about  A,  do  not  equal  the  sum  of  the  moments  of  those 
which  tend  to  turn  it  the  other  way,  then  a  pressure  being 
applied  to  the  system,  equal  and  opposite  to  the  resultant  R, 
will  bring  about  the  equality  of  these  two  sums,  so  that  the 
moment  of  R  must  be  equal  to  the  difference  of  these  sums. 
Let  then  p  equal  the  perpendicular  distance  of  the  direction 
of  E  from  A.  Therefore 


a-f  ....  +mtPw.  .  .  (11), 

in  the  second  member  of  which  equation  the  moments  of 
those  pressures  are  to  be  taken  negatively,  which  tend  to 
communicate  motion  round  A  towards  the  left. 
Dividing  both  sides  by  E-  we  have 


Thus  then  by  equations  (9),  (10),  (12),  the  magnitude  of 
the  resultant  R,  its  inclination  to  the  given  axis  Aa?,  and  the 
perpendicular  distance  of  its  direction  from  the  point  A,  are 
known;  and  thus  the  resultant  pressure  is  completely  deter- 
mined in  magnitude  and  direction. 

THE  PARALLELOPIPEDON  OF  PRESSURES. 

12.  Three  pressures,  Pl?  PQ,  P3,  being  applied  to  the  same 
point  A,  in  directions  a?A,  ^/A,  0A,  which  are  not  in  the 
same  plane,  it  is  required  to  determine  their  resultant. 

Take  the  lines  P,  A,  P2  A,  P3  A,  to  represent  the  pressures 
Pj,  P2,  P3,  in  magnitude  and  direction. 
Complete  the  parallelopipedoii  RPaP8Pl5 
of  which  APj,  AP2,  AP3,  are  adjacent  edges, 
and  draw  its  diagonal  RA ;  then  will  RA 
represent  the  resultant  of  P,,  P2,  P3,  in 
direction  and  magnitude.  For  since 
PjSPjjA  is  a  parallelogram,  whose  adjacent 
sides  Pj  A,  P2  A,  represent  the  presurea 
Pa  and  Pa  in  magnitude  and  direction,  therefore  its  diagonal 


OF   THREE   PRESSURES.  15 

SA  represents  the  resultant  of  these  two  pressures.  And 
similarly  KA,  the  diagonal  of  the  parallelogram  KSAP3,  re- 
presents in  magnitude  and  direction  the  resultant  of  SA  and 
P8,  that  is,  of  F15  P2  and  P3,  since  SA  is  the  resultant  of 
P,  and  P9. 

It  is  evident  that  the  fourth  pressure  necessary  to  produce 
an  equilibrium  with  Pn  P9,  P8,  being  equal  and  opposite  to 
their  resultant,  is  represented  in  magnitude  and  direction 
by  AR. 

13.  Three  pressures,  P,,  P2,  P8,  "being  in  equilibrium,  it  is 
required  to  determine  the  third  P3  in  terms  of  the  other 
two,  and  their  inclination  to  one  another. 

Let  APj  and  APa  represent  the  pressures  Pl  and  P2  in 
magnitude  and  direction,  and  let  the  inclination 
P,  AP,  of  JP1  to  P2  be  represented  by  A-  Com- 
plete the  parallelogram  APX  RP2,  and  draw  its 
diagonal  AR.  Then  does  AR  represent  the 
resultant  of  P,  and  P2  in  magnitude  and  direc- 
tion. But  this  resultant  is  in  equilibrium  with  P3,  since  P, 
and  P2  are  in  equilibrium  with  P3.  It  acts,  therefore,  in  the 
same  straight  line  with  P3,  but  in  an  opposite  direction,  and 
is  equal  to  it.  Since  then  AR  represents  this  resultant  in 
magnitude  and  direction,  therefore  RA  represents  P3  in  mag- 
nitude and  direction. 


Now,  A]?=AP>—  2AFX  .  PJK  .  cos. 
also,  AP.R^Tr—  ^AP^Tr—  A,  P1R=AP,,  and  AP15  AP2) 
AR,  represent  P,,  P2,  P3,  in  magnitude. 

/.     ?/  =  ?,-—  2?^.  COB.  (7T—  A)  +P,-). 
NOW  COS.  (7T—  A)  =  —COS.  A,    .'.  P.'zrP^  +  SP,?,  COS.  A 

(13). 


14.  If  three  pressures,  P1?  P2,  P3,  he  in  equilibrium,  any  two 
of  them  are  to  one,  another  inversely  as  the  sines  of  their 
inclinations  to  the  third. 

Let  the  inclination  of  Pj  to  P3  be  represented  by  A>  an(i 
that  of  Pa  to  P,  by  A- 


rrrTT  —  P.AP.rrzT—A,        "    SlU.  P.ARrzrsin.  &  ' 

P1RA=P,AR=^—  P2AP3=rr—  A,     /.  sin.  P1KA=Sin.  A- 


16  OF   PARALLEL   PRESSURES. 

AF,      AP,      sin.  P,KA 


Also, 


AP2  ~~  P,R  ~  sin. 


sn. 


That  is,  P,  is  to  P2  inversely,  as  the  sine  of  the  inclina- 
tion of  PJ  to  P3  is  to  the  sine  of  the  inclination  of  P2  to  P3. 
Therefore,  &c.  &c.  [Q.  E.  D.] 

OF  PARALLEL  PRESSURES. 

15.  The  principle  of  the  equality  of  moments  obtains  in 
respect  to  pressures  in  the  same  plane  whatever  may  be 
their  inclinations  to  one  another,  and  therefore  if  their 
inclinations  be  infinitely  small,  or  if  they  he  parallel. 

In  this  case  of  parallel  pressures,  the  same  line  AB,  which 
3  is  drawn  from  a  given  point  A,  perpendicular 
to  one  of  these  pressures,  is  also  perpendicular 
to  all  the  rest,  so  that  the  perpendiculars  are 
°here  the  parts  of  this  line  AM1?  AM2,  &c. 
intercepted  between  the  point  A  and  the  direc- 
tions of  the  pressures  respectively.  The  principle  is  not  how- 
ever in  this  case  true  only  in  respect  to  the  intercepted  parts 
of  this  perpendicular  line  AB,  but  in  respect  to  the  inter- 
cepted parts  of  any  line  AC,  drawn  through  the  point  A 
across  the  directions  of  the  pressures,  since  the  intercepted 
parts  Amx,  Am2,  Am3,  &c.  of  this  second  line  are  proportional 
to  those,  AM15  AM2,  &c.  of  the  first. 

Thus  taking  the  case  represented  in  the  figure,  since  by 
the  principle  of  the  equality  of  moments  we  have, 


AM,  .  P,  +  AM4  .  P4=AM2  . 
dividing  both  sides  by  AM6, 


.  -  , 

AM.  '  r'+  AM5  '  r*~~  AM5  '  r'+  AM5  ' 

AM,     Am,    AM2     Am2 
Butbysimilartriangles,  = 


'  Am.  '          Am,  '     4~~AmB  '          Am§  ' 
Therefore  multiplying  by  Am6, 

Am,  .  P.+  Amt  .  P4  =Am^  .  P3  +  Am,  .  P8+  Am~  .  P6. 
Therefore,  &c.  [Q.B.D.] 


OF   PARALLEL   PRESSURES.  17 


16.  To  find  the  resultant  of  any  number  of  parallel  pressures 
in  the  same  plane. 

It  is  evident  that  if  a  pressure  equal  and  opposite  to  the 
resultant  were  added  to  the  system,  the  whole  would  be  in 
equilibrium.  And  being  in  equilibrium  it  has  been  shown 
(Art.  8.),  that  if  the  pressures  were  all  moved  from  their 
present  points  of  application,  so  as  to  remain  parallel  to  their 
existing  directions,  and  applied  to  the  same  point,  they  are 
such  as  would  be  in  equilibrium  about  that  point.  '  But 
being  thus  moved,  these  parallel  pressures  would  all  have 
their  directions  in  the  same  straight  line.  Acting  therefore  all 
in  the  same  straight  line,  and  being  in  equilibrium,  the  sum 
of  those  pressures  whose  tendency  is  in  one  direction  along 
that  line  must  equal  the  sum  of  those  whose  tendency  is  in 
the  opposite  direction.  Now  one  of  these  sums  incluaes  the 
resultant  R.  It  is  evident  then  that  before  R  was  introduced 
the  two  sums  must  have  been  unequal,  and  that  R  equals,  the 
excess  of  the  greater  sum  over  the  less ;  and  generally  that  if 
2P  represent  the  sum  of  any  number  of  parallel  pressures, . 
those  whose  tendency  is  in  one  direction  being  taken  with: 
the  positive  sign,  and  those  whose  tendency  is  in  the  opposite 
direction,  with  the  negative  sign ;  then 

R  =  2P (15). 

the  sign  of  R  indicating  whether  it  act  in  the  direction  of 
those  pressures  which  are  taken  positively,  or  those  which  are 
taken  negatively. 

Moreover  since  these  pressures,  including  R,  are  in  equi- 
librium, therefore  the  sum  of  the  moments  about  any  point, 
of  those  whose  tendency  is  to  communicate  motion  in  one 
direction,  must  equal  the  sum  of  the  moments  of  the  rest — 
these  moments  being  measured  on  any  line,  as  AC ;  but  one 
of  these  sums  includes  the  moment  of  R ;  these 
two  sums  must  therefore,  before  the  introduc- 
tion of  R,  have  been  unequal,  and  the  moment 
of  R  must  be  equal  to  the  excess  of  the  greater 
sum  over  the  less,  so  that,  representing  the 
sum  of  the  moments  of  the  pressures  (R  not  being  included) 
by  2  m*  P,  those  whose  tendency  is  to  communicate  motion 
in  one  direction,  having  the  positive  sign,  and  the  rest  the 
negative  ;  and  representing  by  x  the  ^  distance  from  A,  mea- 
sured along  the  line  AC,  at  which  R  intersects  that  line,  we 
have,  since  xR  is  the  moment  of  R,  xR  =  2  ml  P,  where  the 


18  OF   PARALLEL   PKESSURES. 

sign  of  a?R  indicates  the  direction  in  which  R  tends  to  turn 
the  system  about  A,  but  E-  =  2P, 

.  (16). 


2P 

Equations  (15)  and  (16)  determine  completely  the  magni- 
tude and  the  direction  of  the  resultant  of  a  system  of  parallel 
pressures  in  the  same  plane. 

IT.  To  determine  tlw  resultant  of  any  number  of  parallel 
pressures  not  in  the  same  plane. 

Let  Pj  'and  P2  be  the  points  of  application  of  any  two  of 
these  pressures,  and  let  the  pressures  themselves 
be  represented  by  P1  and  P2.  Also  let  their 
resultant  Rj  intersect  the  line  joining  the  points 
PI  and  P2  in  the  point  Rj  ;  produce  the  line 
P1?  Pa,  to  intersect  any  plane  given  in  position, 
in  the  point  L.  Through  the  points  P15  P2,  and  R15  draw 
P.Mj,  PaM2,  and  Rj]^  perpendicularly  to  this  plane:  these 
lines  will  be  in  the  same  plane  with  one  another  and  with 
Pj  L  ;  let  the  intersection  of  this  last  mentioned  plane  with 
the  first  be  LM,,  then  will  PjMj,  P2M2,  and  R^  be  per- 
pendiculars to  LM:  ;  moreover  by  the  last  proposition, 


But  by  similar  triangles, 

LP_PMl  LP2_P21VI2 


Let  now  the  resultant,  R2,  of  R:  and  Ps 
intersect  the  line  joining  the  points  R,  and 
P3  in  the  point  R2,  and  similarly  let  the 
resultant,  R3,  of  R2  and  P4  intersect  the 
me  j°mmg  the  points  R2  and  P4  in  the 
point  Rg,  and  so  on  :  then  by  the  last  equa- 
tion. 


OF    PARALLEL    PRESSURES. 


Similarly,       E, .  R      +  P3 .  P.M.  =  E2 


&c.      +       &c.      =        &c. 
E 


7i_2 


Adding  these  equations,  and  striking  out  terms  common  to 
both  sides, 


P,  .  P       +  P2P       +  .  .  .  +  P.  ."P3L  =R»-1  . 


P2M2+   .....   +PW  .  PnMn; 


.    p N-       -'     > 

in  which  expression  those  of  the  parallel  pressures  P15  P2, 
&c.  which  tend  in  one  direction,  are  to  be  taken  positively, 
whilst  those  which  tend  in  the  opposite  direction  are  to  be 
taken  negatively. 

The  line  En_i  N»_i  represents  the  perpendicular  distance 
from  the  given  plane  of  a  point  through  which  the  resultant 
of  all  the  pressures  P1?  P2  .  .  .  .  Pn,  passes.  In  the  same 
manner  may  be  determined  the  distance  of  this  point  from 
any  other  plane.  Let  this  distance  be  thus  determined  in 
respect  to  three  given  planes  at  right  angles  to  one  another. 
Its  actual  position  in  space  will  then  be  known.  Thus  then 
we  shall  know  a  point  through  which  the  resultant  of  all  the 
pressures  passes,  also  the  direction  of  that  resultant,  for  it  is 
parallel  to  the  common  direction  of  all  the  pressures,  and  we 
shall  know  its  amount,  for  it  is  equal  to  the  sum  of  all  the 
pressures  with  their  proper  signs.  Thus  then  the  resultant 
pressure  will  be  completely  known.  The  point  E^i  is  called 
the  CENTRE  OF  PARALLEL  PRESSURES. 

18.  The  product  of  any  pressure  by  its  perpendicular  dis- 
tance from  a  plane  (or  rather  the  product  of  the  number  of 
units  in  the  pressure  by  the  number  of  units  in  the  perpen- 
dicular), is  called  the  moment  of  the  pressure,  in  respect  to 
that  plane.  Whence  it  follows  from  equation  (17)  that  the 
sum  of  the  moments  of  any  number  of  parallel  pressures  in 


20  THE   CENTRE   OF   GRAVITY. 

respect  to  a  given  plane  is  equal  to  the  moment  of  their 
resultant  in  respect  to  that  plane. 

19.  It  is  evident,  from  equation  (18),  that  the  distance 
~Nn—i  of  the  centre  of  pressure  of  any  number  of 
parallel  pressures  from  a  given  plane,  is  independent  of  the 
directions  of  these  parallel  pressures,  and  is  dependent 
wholly  upon  their  amounts  and  the  perpendicular  distances 
PjM^  P2M2,  &c.  of  their  points  of  application  from  the 
given  plane. 

So  that  if  the  directions  of  the  pressures  were  changed, 
provided  that  their  amounts  and  points  of  application 
remained  the  same,  their  centre  of  pressure,  determined  as 
above,  would  remain  unchanged;  that  is,  the  resultant, 
although  it  would  alter  its  direction  with  the  directions  of 
the  component  pressures,  would,  nevertheless,  always  pass 
through  the  same  point. 

The  weights  of  any  number  of  different  bodies  or  different 
parts  of  the  same  body,  constitute  a  system  of  parallel  pres- 
sures ;  the  direction,  therefore,  through  this  system  of  the 
resultant  weight  may  be  determined  by  the  preceding  pro- 
position ;  their  centre  of  pressure  is  their  centre  of  gravity. 

THE  CENTRE  OF  GRAVITY. 

20.  The  resultant  of  the  weights  of  any  number  of  bodies 
or  parts  of  the  same  body  unitea  into  a  system  of  inva- 
riable form  passes  through  the  same  point  in  it,  into  what- 
ever position  it  may  be  turned. 

For  the  effect  of  turning  it  into  different  positions  is  to 
cause  the  directions  of  the  weights  of  its  parts  to  traverse 
the  heavy  body  or  system  in  different  directions,  at  one  time 
lengthwise  for  instance,  at  another  across,  at  another 
obliquely  •  and  the  effect  upon  the  direction  of  the  resultant 
weight  through  the  body,  produced  by  thus  turning  it  into 
different  positions,  and  thereby  changing  the  directions  in 
which  the  weights  of  its  component  parts  traverse  its  mass, 
is  manifestly  the  same  as  would  be  produced,  if  without  alter- 
ing the  position  of  the  body,  the  direction  of  gravity  could 
be  changed  so  as,  for  instance,  to  make  it  at  one  time  tra- 
verse that  body  longitudinally,  at  another  obliquely,  at  a 
third  transversely.  But  by  Article  19,  this  last  mentioned 
change,  altering  the  common  direction  of  the  parallel  pres- 


THE   CENTRE   OF   GRAVITY.  21 

sures  through  the  body  without  altering  their  amounts  or 
their  points  of  application,  would  not  alter  the  position  of 
their  centre  of  pressure  in  the  body ;  therefore,  neither  would 
the  first  mentioned  change.  Whence  it  follows  that  the 
centre  of  pressure  of  the  weights  of  the  parts  of  a  heavy 
body,  or  of  a  system  of  invariable  form,  does  not  alter  its 
position  in  the  body,  whatever  may  be  the  position  into 
which  the  body  is  turned;  or  in  other  words,  that  the 
resultant  of  the  weights  of  its  parts  passes  always  through 
the  same  point  in  the  body  or  system  in  whatever  position 
it  may  be  placed. 

This  point,  through  which  the  resultant  of  the  weights  of 
the  parts  of  a  body,  or  system  of  bodies  of  invariable  form, 
passes,  in  whatever  position  it  is  placed ;  or,  if  it  be  a  body 
or  system  of  variable  form,  through  which  the  resultant 
would  pass,  in  whatever  position  it  were  placed,  if  it  became 
rigid  or  invariable  in  its  form,  is  called  the  CENTRE  OF 
GRAVITY. 

21.  Since  the  weights  of  the  parts  of  a  body  act  in 
parallel  directions,  and  all  tend  in  the  same  direction,  there- 
fore their  resultant  is  equal  to  their  sum.  Now,  the  result- 
ant of  the  weights  of  the  parts  of  the  body  would  produce, 
singly,  the  same  effect  as  it  regards  the  conditions  of  the 
equilibrium  of  the  body,  that  the  weights  of  its  parts 
actually  do  collectively,  and  this  weight  is  equal  to  the  sum 
of  the  weights  of  the  parts,  that  is,  to  the  whole  weight  of 
the  body,  and  in  every  position  it  acts  vertically  downwards 
through  the  same  point  in  the  body,  viz.  the  centre  of 
gravity.  Thus  then  it  follows,  that  in  every  position  of  the 
body  and  under  every  circumstance,  the  weights  of  its  parts 
produce  the  same  effect  in  respect  to  the  conditions  of  its 
equilibrium,  as  though  they  were  all  collected  in  and  acted 
through  that  one  point  of  it — its  centre  of  gravity* 

*  That  the  resultant  of  the  weights  of  all  the  parts  of  a  rigid  body  passes 
in  all  the  positions  of  that  body  through  the  same  point  in  it  is  a  property  of 
many  and  most  important  uses  in  the  mechanism  of  the  universe,  as  well  as  in 
the  practice  of  the  arts ;  another  proof  of  it  is  therefore  subjoined,  which 
may  be  more  satisfactory  to  some  readers  than  that  given  in  the  text.  The 
system  being  rigid,  the  distance  PI,  P2,  of  the  points  of 
application  of  any  two  of  the  pressures  remains  the 
same,  into  whatever  position  the  body  may  be  turned : 
the  only  difference  produced  in  the  circumstance  under 
which  they  are  applied  is  an  alteration  in  the  inclina- 
tions of  these  pressures  to  the  line  PI,  P2 :  now  being 
weights,  the  directions  of  these  pressures  always  remain 
parallel  to  one  another,  whatever  may  be  their  inclina- 
tion ;  thus  then  it  follows  by  the  principle  of  the  equa- 


22  THE   CENTRE   OF   GRAVITY. 


22.  To  determine  the  position  jf  the  centre  o£  gravity  of 
two  weights ',  Px  and  reforming  part  of  a  rigid  system. 

Let  it  be  represented  by  G.     Then  since  the  resultant  of 

T@ ^  Pj  and  PQ  passes  through  G,  we  have  by  equa- 

P*  tion  (16),  taking  Pt  as  the  point  from  which  the 
moments  are  measured, 


P  4-P      P  (1—  P      P  P 
x^j-f  jrt  .  r  avr  —  1  3  .  JT  jiT.,, 

P       P  P 

p  r\  _     2j_^_i^_2. 

ilU  -TVfPT 

whence  the  position  of  G  is  known. 

23.  It  is  required  to  determine  the  centre  of  gravity  of  three 
weights  P15  P2,  P3,  not  in  the  same  straight  line,  and  form- 
ing  part  of  a  rigid  system. 

Find  the  centre  of  gravity  G15  of  Px  and  P2,  as  in  the  last 

proposition.     Suppose  the  weights  P1  and  P2  to 

jft  be  collected  in  G15  and  find  as  before  the  com- 

^^Jc,  mon  centre  of  gravity  G2  of  this  weight  Pj  +  P,, 

r.*--  —  Vj     so  collected  in  Ga,  and  the  third  weight  P3.     It 

Lg    is  evident  that  this  point  G2  is  the  centre  of 

gravity  required.      Since   G2  is  the    centre  of 

gravity  of  P3  and  P^  P2  collected  in  G15  we  have  by  the 

last  proposition 

G?,  .  P., 


G.P..F. 


lity  of  moments  (Art.  15),  that  Pi4-P2  •  PiRi—  P2 .  PiP2,  so  that  for  every 
such  inclination  of  the  pressures  to  PI  P2,  the  line  PI  HI  is  of  the  same  length, 
and  the  point  Rj  therefore  the  same  point ;  therefore,  the  line  P3Ri  is  always 
the  same  line  in  the  body;  and  RI  which  equals  P!-f-P2,  is  always  the  same 
pressure,  as  also  is  P8,  and  these  pressures  always  remain  parallel,  therefore, 
for  the  same  reason  as  before,  R2  is  always  the  same  point  in  the  body  in 

whatever  position  it  may  be  turned,  and  so  of  R3,  R4 and  R.-I.     That 

is,  in  every  position  of  the  body,  the  resultant  of  the  weights  of  its  parts 
passes  through  the  same  point  R«-i  in  it.  Since  the  resultant  of  the  weights 
of  the  parts  of  a  body  always  passes  through  its  centre  of  gravity,  it  is 
evident, 'that  a  single  force  applied  at  that  point  equal  and  opposite  to  this 
resultant,  that  is,  equal  in  amount  to  the  whole  weight  of  the  body,  and  in  a 
direction  vertically  upwards,  would  in  every  position  of  the  body  sustain  it. 
This  property  of  the  centre  of  gravity,  viz.  that  it  is  a  point  in  the  body  where 
a  single  force  would  support  it  is  sometimes  taken  as  the'  definition  of  it. 


OF   A   TRIANGLE.  23 

If  P15  P2,  P3,  be  all  equal,  then 

%<M= 

Moreover  in  this  case, 


24.  To  find  the  centre  of  gravity  of  four  weights  not  in  thf> 

same  plane. 

Let  P1?  P2,  P3,  P4,   represent  these   weights;    find  the 

centre  of  gravity  G2  of  the  weights  P,,  P3, 

y5_  P3,  as  in  the  last  proposition  ;  suppose  these 

/j\  three  weights  to  he  collected  in  G2,  and  then 

//I    \  find  the  centre  of  gravity  G3  of  the  weight 

/iift...  \        tnus  collected  in  G-2  and  t\.     G3  will  he  the 

jgjSfc^3^      centre  of  gravity  required,  and  since  G3  is 

the  centre  of  gravity  of  P4  acting  at  the 

point  P4,  and  of  P^Pa+P,  collected  at  G2, 


If  all  these  weights  be  equal,  then  by  the  above  equation, 


also,          _G1G?=i_G1P3, 
and  &?!=$  P.P.- 


25.    THE   CENTRE   OF   GRAVITY   OF   A   TRIANGLE. 

Let  the  sides  AB  and  EC  of  the  triangular  lamina  ABC 
be  bisected  in  E  and  D,  and  the  lines  CE  and 
AD  drawn  to  the  opposite  angles,  then  is  the 
intersection  G  of  these  lines  the  centre  of  gravity 
of  the  triangle  :  for  the  triangle  may  be  supposed 
to  be  made  up  of  exceedingly  narrow  rectangular 
strips  or  bands,  parallel  to  JBC,  each  of  which  will 
be  bisected  by  the  line  AD;  for  by  similar  triangles 
PK  :  DB  ::  AE  :  AD  ::  KQ  :  DC,  therefore,  alternando, 
PK  :  KQ::DB  :  DC;  but  DB=DC;  therefore  PR=PvQ. 

Therefore,  each  of  the  elementary  bands,  or  rectangles 
parallel  to  BC,  which  compose  the  triangle  ABC,  would 
separately  balance  on  the  line  AD ;  therefore,  all  of  them 


24  THE   CENTRE   OF   GRAVITY 

joined  together  would  balance  on  the  line  AD,  therefore  the 
centre  of  gravity  of  the  triangle  is  in  AD. 

In  the  same  manner  it  may  be  shown  that  the  centre  of 
gravity  of  the  triangle  is  in  the  line  CE ;  therefore,  the  cen- 
tre of  gravity  is  at  the  intersection  G  of  these  lines. 

Now  DG=J  DA :  for  imagine  the  triangle  to  be  without 
weight,  and  three  equal  weights  to  be  placed  at  the  angles 
A,  6,  and  C,  then  it  is  evident  that  these  three  weights  will 
balance  upon  AD ;  for  AD  being  supported,  the  weight  A 
will  be  supported,  since  it  is  in  that  line ;  moreover,  B  and 
C  will  be  supported  since  they  are  equidistant  from  that 
line. 

Since,  then,  all  three  of  the  weights  will  balance  upon 
AD,  their  centre  of  gravity  is  in  AD.  In  like  manner  it 
may  be  shown  that  the  centre  of  gravity  of  all  three  weights 
is  in  CE  ;  therefore  it  is  in  G,  and  coincides  with  the  centre 
of  gravity  of  the  triangle. 

]N  ow,  suppose  the  weights  B  and  C  to  be  collected  in  their 
centre  of  gravity  D,  and  suppose  each  weight  to  be  repre- 
sented in  amount  by  A,  a  weight  equal  to  2A  will  then  be 
collected  in  D,  and  a  weight  equal  to  A  at  A,  and  the  centre 
of  gravity  of  these  is  in  G ;  therefore  DA  x  A  =  DG  x 
(2A  +  A), 

.  • .  D  A  =  3  DG,  or  DG  =  £  DA.*  [Q.E.D.] 

26.    THE   CENTKE   OF    GRAVITY   OF   THE   PYRAMID. 

Let  ABC  be  a  pyramid,  and  suppose  it  to  be 
made  up  of  elementary  laminae  l)cd,  parallel  to 
the  base  BCD.     Take  G,  the  centre  of  gravity 
of  the  base  BCD,  and  join  AG;   then  AG  will 
pass   through  the  centre  of  gravity  g  of  the 
lamina  Icd^  therefore  each  of  the  laminae  will  separately 
balance  on  the  straight  line  AG ;  therefore  the  laminae  when 
combined  will  balance  upon  this  line ;  therefore  the  whole 
figure  will  balance  on  AG,  and  the  centre  of  gravity  of 
the  whole  is  in  AG.    In  like  manner  if  the  centre  of  gravity 
H  of  the  face  ABD  be  taken,  and  CH  be  joined,  then  it  may 
be  shown  that  the  centre  of  gravity  of  the  whole  is  in  Cli ; 

« 

*  Note  (g)  Ed.  App. 

f  For  produce  the  plane  ABG-  to  intersect  the  plane  ADC  in  AM,  then  by 
similar  triangles  DM  :  MC  : :  dm  :  me,  but  DM  =  MC ;  therefore  dm  =  me.  Also 
by  similar  triangles  GM  :  BM::#m  :  bm,  but  GM  =  i  BM;  therefore  gm  =  $ 
bm.  Since  then  dm  =  J  dc  and  gm  =  $•  bm,  therefore  g  is  the  centre  of  gravity 
of  the  triangle  bdc. 


OF   A   PYRAMID.  25 

therefore  the  lines  AGr  and  CII  intersect,  and  the  centre  of 
gravity  is  at  their  intersection  K. 

Now  GK  is  one-fourth  of  GA ;  for  suppose  equal  weights 
to  be  placed  at  the  angles  A,  B,  C,  and  D  of  the  pyramid 
(the  pyramid  itself  being  imagined  without  weight),  then 
will  these  four ^ weights  balance  upon  the  line  AG,  for  one 
of  them,  A,  is  in  that  line,  and  the  line  passes  through  the 
centre  of  gravity  G  of  the  other  three. 

Since,  then,  the  equal  weights  A,  B,  C,  and  D  balance 
upon  the  line  AG,  their  centre  of  gravity  is  in  AG ;  in  the 
same  manner  it  may  be  shown  that  the  centre  of  gravity  of 
the  four  weights  is  in  CH,  therefore  it  is  in  K,  and  coincides 
with  the  centre  of  gravity  of  the  pyramid. 

Now  let  the  number  of  units  in  each  weight  be  repre- 
sented by  A,  and  let  the  three  weights  B,  C,  and  D  be 
supposed  to  be  collected  in  their  centre  of  gravity  G ;  the 
four  weights  will  then  be  reduced  to  two,  viz.  3A  at  G,  and 
A  at  A,  whose  common  centre  of  gravity  is  K, 

/.  GKx3A+A  =  GAxA, 

/.  4GK  =  GA  or  GK  =  J  GA.*  [Q.E.D.] 

27.  The  centre  of  gravity  of  a  pyramid  with  a  polygonal  lase 
is  situated  at  a  vertical  height  from  the  base,  equal  to  one 
fourth  the  whole  height  of  the  pyramid. 

For  any  such  pyramid  ABCDEF  may  be  supposed  to 
be  made  up  of  triangular  pyramids  ABOF, 
A  CDF,  and  ADEF,  whose  centres  of  gravity 
G,  H,  and  K,  are  situated  in  lines  AL,  AM, 
and  AN,  drawn  to  the  centres  of  gravity  L,  M, 
and  N  of  their  bases' ;  LG  being  one-fourth  of 
LA,  Mil  one-fourth  of  MA,  and  NK  one-fourth 
of  NA.  The  points  G,  H,  and  K,  are  therefore  in  a  plane 
parallel  to  the  base  of  the  pyramid,  and  whose  vertical  dis- 
tance from  the  base  equals  one-fourth  the  vertical  height  of 
the  pyramid. 

Since  then  the  centres  of  gravity  G,  H,  and  K  of  the  ele- 
mentary triangular  pyramids  which  compose  the  whole  poly- 
gonal pyramid  are  in  this  plane,  therefore  the  centre  of  gravity 
of  the  whole  is  in  this  plane,  i.  e.  the  centre  of  gravity  of  the 
whole  polygonal  pyramid  is  situated  at  a  vertical  height  from 
the  base,  equal  to  one  fourth  the  vertical  height  of  the  whole 

*  Note  (h)  Ed.  App. 


26  THE   CENTEE   OF   GRAVITY 

pyramid,  or  at  a  vertical  depth  from  the  vertex,  equal  to  three 
fourths  of  the  whole.  JSTow  the  above  proportion  is  true, 
whatever  be  the  number  of  the  sides  of  the  polygonal  base, 
and  therefore  if  they  be  infinite  in  number ;  and  therefore  it 
is  true  of  the  cone,  which  may  be  considered  a  pyramid  hav- 
ing a  polygonal  base,  of  an  infinite  number  of  sides  ;  and  it 
is  true  whether  the  cone  or  pyramid  be  an  oblique  or  a  right 
cone  or  pyramid. 

28.  If  a  body  be  of  a  prismatic  form,  and  symmetrical 
about  a  certain  plane,  then  its  whole  weight  may  be  sup- 
posed to  be  collected  in  the  surface  of  that  plane,  and  uni- 
formly distributed  through  it.  For  let 
ACBEFD  represent  such  a  prismatic 
body,  and  dbc  a  plane  about  which  it  is 
symmetrical :  take  m,  an  element  of  uni- 
form thickness  whose  sides  are  parallel  to 
B  the  sides  of  the  prism,  and  which  is 
terminated  by  the  faces  ACB  and  DFE  of  the  prism  ; 
it  is  evident  that  this  element  m  will  be  bisected  by  the 
plane  abc,  and  that  its  centre  of  gravity  will  therefore 
lie  in  that  plane,  so  that  its  whole  weight  nlay  be  sup- 
posed collected  in  that  plane ;  and  this  being  true  of 
every  other  similar  element,  and  all  these  elements  be- 
ing equal,  it  follows  that  the  whole  weight  of  the  body 
may  be  supposed  to  be  collected  in  and  uniformly  dis- 
tributed through  that  plane.  It  is  in  this  sense  only  that  we 
can  speak  with  accuracy  of  the  weight  and  the  centre  of  gra- 
vity of  a  plane,  whereas  a  plane  being  a  surface  only,  and 
having  no  thickness,  can  have  no  weight,  and  therefore  no 
centre  of  gravity.  In  like  manner  when  we  speak  of  the 
centre  of  gravity  of  a  curved  surface,  we  mean  the  centre  of 
gravity  of  a  body,  the  weights  of  all  whose  parts  may  be  sup- 
posed to  be  collected  and  uniformly  distributed  throughout 
that  curved  surface.  It  is  evident  that  this  condition  is 
approached  to  whenever  the  body  being  hollow,  its  material 
is  exceedingly  thin.  Its  whole  weight  may  then  be  conceived 
to  be  collected  in  a  surface  equidistant  from  its  two  external 
surfaces.  In  like  manner  an  exceedingly  thin  uniform  curved 
rod  may  be  imagined  to  have  its  weight  collected  uniformly 
in  a  line  passing  along  the  centre  of  its  thickness,  and  in  this 
sense  we  may  speak  of  the  centre  of  gravity  of  a  line, 
although  a  line  having  no  breadth  or  thickness  can  have  no 
weight,  and  therefore  no  centre  of  gravity. 


OF   ANY    QUADRILATERAL   FIGURE.  27 


29.    THE   CENTRE   OF    GRAVITY    OF   A   TRAPEZOID. 

Let  AD  and  BO  be  the  parallel  sides  of  the  trapezoid,  of 
B  which  AD  is  the  less.  Let  AD  be  represented 
by  0,  BC  by  5,  and  the  perpendicular  distance 

~  ' 


tlie  tw°  sides  by  ^*  •Draw  DE  parallel 
to  AB.  Let  Gx  be  the  intersection  of 
t]le  ^iag0nals  of  the  parallelogram  ABED, 
then  will4Gt  be  the  centre  of  gravity  of  that  parallelo- 
gram. Bisect  OE  in  L,  join  DL,  and  take  DG2=f  DL, 
then  will  G3  be  the  centre  of  gravity  of  the  triangle  DEC. 
Draw  GjM,  and  G2M2  perpendiculars  to  AD  ;  then  since 
AG,=i  AE,  therefore  G.M^i  FE=4  h.  And  since 
DG2  =  f  DL,  therefore  G2M2  =  |  NL  =  f  h.  Suppose  the 
whole  parallelogram  to  be  collected  in  its  centre  of  gravity 
G1?  and  the  whole  triangle  in  its  centre  of  gravity  Ga.  Let 
G  be  the  centre  of  gravity  of  the  whole  trapezoid,  and  draw 
GM  perpendicular  to  AD.  Then  would  the  whole  be  sup- 
ported by  a  single  force  equal  to  the  weight  of  the  trapezoid 
acting  upwards  at  G.  Therefore  (Art.  17), 


MG  .  ABCD=~G^M1  .  ABED  +  G~M2  .  "CED 

Now,  ABCD  =  i  h  (a  +  5),  ABED  =  ha, 

CED  =  i  A  (7,_a),  G.M,  =  J  A,  G2M2  =  f  A, 
.-.  MG'  .  -J-  h  (a+~b}  =  %  h  .  ha+%  h  .  \  li(b—a\ 
.-.  MG  (a+6)  =  Aa+|  h  (I—  a)  =  -J  h 


(19). 


30.    THE   CENTRE   OF    GRAVITY    OF   ANY   QUADRILATERAL    FIGURE. 

Draw  the  diagonals  AC  and  BD  of  any  quadrilateral  figure 
ABCD,  and  let  them  intersect  in  E, 
and  from  the  greater  of  the  two  parts, 
BE  and  DE,  of  either  diagonal  BD  set 
off  a  part  BF  equal  to  the  less  part. 
Bisect  the  other  diagonal  AC  in  H,  join 
HF  and  take  HG  equal  to  one  third  of 

HF  ;   then  w^ill  G  be  the  centre  of  gravity  of  the  whole 

figure. 

For  if  not,  let  g  be  the  centre  of  gravity,  join  HB  and  HD 

and  take  HGX  —  \  HB  and  HG2  =  i  HD,  then  will  Gx  and 

G2  be  the  centres  of  gravity  of  the  triangles  ABC  and  ADC 


28  THE   CENTRE   OF   GRAVITY. 

respectively  (Art.  25).  Suppose  these  triangles  to  be  col- 
lected in  their  centres  of  gravity  G15  G2 ;  it  is  evident  that 
the  centre  of  gravity  ^,  of  the  whole  figure,  will  be  in  the 
straight  line  joining  the  points  Gx  G2 :  let  this  line  intersect 
AC  in  K ;  then  since  a  pressure  equal  to  the  weight  of  the 
whole  figure  acting  upwards  at  </,  will  be  in  equilibrium  with 
the  weights  of  the  triangles  collected  in  Gj  and  G2,  we  have, 
by  the  principle  of  the  equality  of  moments  (Art.  15), 


E^  .  ABCD^KG,  .  ABC—  KG2  .  ADC. 

Now  since  HG,  =  \  HB,  and  HG2  =  \  HD,  therefore  G,  G2 
is  parallel  to  DB,  therefore  KG^i  BE,  and  KGa  =  J  DE. 
Now  let  the  angle  AED  =  BEC  —  i.  Therefore  the  perpen- 
dicular from  B  upon  AC  —  BE  sin.  «,  and  that  from  33  =  DE 

sin.  «,  therefore  area  of  triangle  ABC  —  %  AC  .  BE  sin.  «, 
and  area  of  triangle  ADC  =  J  AC  .  DE  sin.  i,  therefore  area 
of  quadrilateral  ABGD  =  j  AC  .  BE~  sin.  i+J  AC  .  DE 

sin.  »  =  J  (BE  +  DE)  AC  sin.  *.  Substituting  these  values  in 
the  preceding  equation, 

!Lg  .  J  (BE+DE)J[C  sin^iJBE  .  J  AC  .  BE  sin.  i  — 
DE  .      AC  .  DE  sin.  i 


(BE+DE)  =  -i-  (BF—  DE5), 


But  sin-ce  HG  =  i  HF,  .\KG  =  ^  FE,  .-.E^  =  KG;  that 
is,  the  true  centre  of  gravity  g  coincides  with  the  point  G. 
Therefore,  &c.  [Q.E.D.] 

*31.  In  the  examples  hitherto  given,  the  centre  of  pressure 
of  a  system  of  weights,  or  their  centre  of  gravity,  lias  been 
determined  by  methods  which  are  indirect  as  compared  with 
the  direct  and  general  method  indicated  in  Article  IT.  That 
method  supposes,  however,  a  determination  of  the  sum  of  the 
moments  of  the  weights  of  all  the  various  elements  of  the 
body  in  respect  to  three  given  planes.  Now  in  a  continuous 
body  these  elements  are  infinite  in  number,  each  being  infi- 
nitely small  ;  this  determination  supposes,  therefore,  the  sum- 
mation of  an  infinite  number  of  infinitely  small  quantities, 
and  requires  an  application  of  the  principles  of  the  intregal 
calculus. 

Let  AM  be  taken  to  represent  any  small  element  of  the 


THE    CENTRE    OP    GRAVITY. 


29 


volume  M  of  a  continuous  body,  and  x  its  perpendicular 
distance  from  a  given  plane.  Then  will  OJJA  AM  represent 
the  moment  of  the  weight  of  this  element  about  that  plane, 
fx  representing  the  weight  of  each  unit  of  the  volume  M. 
Let  \&x  AM  represent  the  sum  of  all  such  moments,  taken  in 
respect  to  all  the  small  elements,  such  as  AM,  which  make 
up  the  volume  of  the  body.  Then  if  G^  represent  the  dis- 
tance of  the  centre  of  gravity  of  the  body  from  the  given 
plane  ;  since  i^2a?AM  represents  the  sum  of  the  moments  of  a 
system  of  parallel  pressures  about  that  plane,  f*M  the  sum  of 
those  pressures,  and  Gx  the  distance  of  their  centre  of  pres- 
sure from  the  plane  (Art.  19),  it  follows  by  equation  (18)  that 

4^3^ (20)_ 

Now  it  is  proved  in  the  theory  of  the  integral  calculus,* 
that  a  sum,  such  as  is  represented  by  the  above  expression 
2a?AM,  whose  terms  are  infinite  in  number,  and  each  the  pro- 
duct of  a  finite  quantity  a?,  and  an  infinitely  small  quantity 
AM,  and  in  which  M  is,  as  in  this  case,  a  function  of  x  (and 
therefore  x  a  function  of  M),  is  equal  to  the  definite  integral 


/  a 


Therefore,  generally, 


fx 


M 


Similarly, 


J'ydM. 


(21). 


M 

In  the  two  last  of  which  equations  y  and  s  are  taken  to  repre- 
sent, respectively,  the  distances  of  the  element  AM  of  the 

*  Poisson,  Journal  de  1'Ecole  Polytechnique,  18me  cahier,  p.  320,  or  Art.  2, 
in  the  Treatise  on  Definite  Integrals  in  the  Encyclopaedia  Metropolitana  by  the 
author  of  this  work.  See  Appendix,  note  A. 


30  THE   CENTRE   OF    GRAVITY. 

body  from  two  other  planes,  as  x  represents  its  distance  from 
the  first  plane  ;  and  Gt  and  G2  to  represent  the  distances  of 
its  centre  of  gravity  from  those  planes.  The  distances  G,, 
G2,  G3,  of  the  centre  of  gravity  from  three  different  planes 
being  thus  known,  its  actual  position  in  space  is  fully  deter- 
mined. These  three  planes  are  usually  taken  at  right  angles 
to  one  another,  and  are  then  called  rectangular  co-ordinate 
planes,  and  their  common  intersections  rectangular  co-ordi- 
nate axes. 

If  the  centre  of  gravity  of  the  body  be  known  to  lie  in  a 
certain  plane,  and  one  of  the  co-ordinate  planes  spoken  of 
above,  as  for  instance  that  from  which  G3  is  measured,  be 
taken  to  coincide  with  this  plane  in  which  the  centre  of  gra- 
vity is  known  to  lie,  then  G3  ==  0,  and  the  position  of  the  cen- 
tre of  gravity  is  determined  by  the  two  first  only  of  the  above 
three  equations.  This  case  occurs  when  the  body,  whose 
centre  of  gravity  is  to  be  determined,  is  symmetrical  about  a 
certain  plane,  since  then  its  centre  of  gravity  evidently  lies 
in  its  plane  of  symmetry.  If  the  centre  of  gravity  of  the 
body  be  known  to  lie  in  a  certain  line,  and  two  of  the  co-or- 
dinate planes,  those  for  instance  from  which  G2  and  G3  are 
measured,  be  taken  so  as  to  intersect  one  another  in  that  line, 
then  the  centre  of  gravity  will  be  in  both  those  planes  ;  there- 
fore G2  =  0  and  G3  =  0,  and  its  position  is  determined  by  the 
first  of  the  preceding  equations  alone.  This  case  occurs 
when^  the  body  is  symmetrical  about  a  given  line  ;  its  centre 
of  gravity  is  then  manifestly  in  that  line. 

"*32.  THE  CENTRE  OF  GRAVITY  OF  A  CURVED  LINE  WHICH  LIES 
WHOLLY  IN  THE  SAME  PLANE. 

Taking  M  to  represent  the  length  S  of  such  a  line,  we 
have,  by  equations  (21), 


.-  ,  .. 

EXAMPLE.  —  Let  it  ~be  required  to  determine  the  centre  of 
gravity  of  a  circular  arc  EF. 

The  centre  of  gravity  of  such  an  arc  is  evidently  in  the 
radius  CA,  which  bisects  it;  since  the  arc 
is  symmetrical  about  that  radius.  Take  a 
plane  Cy  perpendicular  to  this  radius,  and 
passing  through  the  centre,  to  measure  the 
moments  from.  Let  x  represent  the  dis- 
tance PM  of  any  point  P  in  this  arc  from 


OF   A   CURVED   LINE.  31 

this  plane ;  also  let  s  represent  the  arc  P  A,  and  S  the  aro 
E  AF,  a  the  radius  C  A,  and  C  the  chord  EF. 


.-.  x  =  PM  =  OF  cos.  CPM  =  CP  cos.  AGP  =  a  cos.  - 
#8  IS 

.'.  f  xdS=  a  I  cos.  —ds=a*  /.cos.  —  d\  —  )—  2&2sin.(  —  )> 
•/  •/  a  v  a     \a/  \af 


the  integral  being  taken  between  the  limits  -JS  and  —  JS, 
because  these  are  the  values  of  s  which  correspond  to  the 
extreme  points  F  and  E  of  the  arc. 

Now  2a  sin.  J  (—  )  =  chord  of  EAF  =  C,  /.Juarffl  =  00, 

"Gi  =  lT  ......  (23). 

The  distance  of  the  centre  of  gravity  of  a  circular  arc  from 
the  centre  of  the  circle  is  therefore  a  fourth  proportional  to 
the  length  of  the  arc,  the  length  of  the  chord,  and  the  radius 
of  the  arc. 

*33.  THE   CENTRE   OF    GRAVITY   OF   A   CURVILINEAR  AREA 
WHICH  LIES  WHOLLY  IN  THE  SAME  PLANE. 

'Let  BAG  represent  such  an  area.  If  x  and  y  represent 
the  perpendicular  distances  PN  and  PM  of  any 
point  P  in  the-  curve  AB  from  planes  AC  and 
AD,  perpendicular  to  the  plane  of  the  given  area 
and  to  one  another,  and  M  represent  the  area 
PAM,  then,  considering  this  area  to  be  made  up 
of  rectangles  parallel  to  PM,  the  width  of  each 
of  which  is  represented  by  the  exceedingly  small  quantity 
Aa?,  the  area  AM  of  each  such  rectangle  will  be  -represented 
by  yAaj,  and  its  moment  about  AD  by  v-xy&x. 


j 


xydx 


Therefore  by  equation  (20),  G,  =  =  -g—  .  .  (24). 

A  similar  expression  determines  the  value  o£  G2  ;  butane 
more  convenient  for  calculation  is  obtained,  if  we  consider 
the  weight  of  each  of  the  rectangles,  whose  length  is  y,  to 
be  collected  in  its  centre  of  gravity,  whose  distance  from  AC 


32  THE   CENTRE   OF   GRAVITY. 

is  \y.     The  moment  of  the  weight  of  each  rectangle  about 
AC  will  then  be  represented  ^y*&x  ;  whence  it  follows  that 

»i 
tfdx 

.....  (25). 


-  2         M 

EXAMPLE. — Suppose  the  curve  APB  to  be  a  parabola,  whose 
axis  is  AC. 

Dl  B      By  the  equation  to  the -parabola  y1  =  4##,  if  a 

]  be  the  distance  of  the  focus  from  the  vertex. 
Moreover,  the  limits  between  which  the  integral 
is  to  be  taken  are  0  and  ajt  and  0  and  y1?  since  at 
A,  x  —  0,  y  =  0,  and  at  C,  x  =  a?1?  y  —  y^ 

therefore  Cxydx  —  2  \/  a  /*x%dx—~  \f  ax1 2 ;  also,  M  =  C  yd® 

£C2  0  0 

051  4  3 

=  2  ya  Cx\dx  —  g  i/tf^f,  therefore  Gj^  ^^. 

o 

®1  «1  ^,4  ,1  _  » .  t 

Also,  jy^dx  —  4^  /xdx=  %ax?  =^ 
J  J  b< 


o 

Q 

therefore  G2  —  Q^. 

o 

If,  then,  G  be  the  centre  of  gravity  of  the  parabolic  area 

ACB,  then  AH  =  ?  AC,     HG  =  -  CB. 

5  -8 

*  34.  THE  CENTRE  OF  GRAVITY  OF  A  SURFACE  OF  REVOLUTION. 

Any  surface  of  revolution  BAC  is  evidently  symmetrical 
about  its  axis  of  revolution  AD,  its  centre  of 
gravity  is  therefore  in  that  axis.  Let  the  mo- 
ments be  measured  from  a  plane  passing  through 
A  and  perpendicular  to  the  axis  AD,  and  let  x 
and  y  be  co-ordinates  of  any  point  P  in  the 
generating  curve  APB  of  the  surface,  and  s  the 
length  of  the  curve  AP.  Then  M  being  taken  to  represent 
the  area  of  the  surface,  and  being  supposed  to  be  made  up 
of  bands  parallel  to  PQ,  the  area  AM  of  each  such  band  is 
represented  (see  Art.  40.)*  by  %ny&s,  and  its  moment  by 


*  Church's  Diff.  Calculus,  Art.  91. 


OF  A  SUKFACE. 


33 


s, 

27T,/ 


(26>. 


EXAMPLE.  —  To  determine  the  centre  of  gravity  of  the 
face  of  any  zone  or  segment  of  a  sphere. 


B»/£/\7        j^  j^ACj  represent  the  surface  of  a  sphere v 
•:,D  whose  centre  is  D,  and  whose  radius  DP  is  repre- 
sented by  #,  and  the  arc  AP  by  s.    Then  x  =  DM 


=  DP  cos. 


sn., 


=#  sn.  -, 
a 


a 

=  %a?  sin.  -  cos.  -  =  a*  sin.  —. 
a        a  a 


S,  S, 

./*  /*       2^ 

.*.  STT  /  xyds  =  TTO?  I  sin.  —  ds 

s,  s, 

=  4  ™°  j  cos.  ^i_  cos.  ^1 


=  TO-  icos.3??  -  cos.3  §il  .....  (27). 
{          «  «  J 

< 

where  S4  and  S2  are  the  values  of  s  at  the  points 
03     B,  and  Ba,  where  the  zone  is  supposed  to  ter- 


minate. 


Also,  since     —  =  27ry,     /.  M  =  tor 


=  27ra  /*sin.  1  <fo  =  2^9   i  cos.  ^-a  -  cos.  ?2  1 
J         a  (         a  a> 


34:  THE   CENTRE   OF   GRAVITY 

...  .(28), 


if  E  be  the  bisection  of  EJE,. 

If  S2  =  0,  or  the  zone  commence  from  A,  then 

G,  =  -a  \  1  +  cos.  §4  =  «cos.2-?i.  .  .  .  (29). 
2      (  a  )  2a 

*35.  THE  CENTRE  OF  GRAVITY  OF  A  SOLID  OF  REVOLUTION. 

Any  solid  of  revolution  BAG  is  evidently  symmetrical 
about  its  axis  of  revolution  AD,  its  centre  of 
gravity  is  therefore  in  that  line  ;  and  taking  a 
plane  passing  through  A  and  perpendicular  to 
that  axis  as  the  plane  from  which  the  moments 
are  measured,  we  have  only  to  determine  the 
distance  AG  of  ^the  centre  of  gravity,  from 
that  plane. 

Now,  if  x  and  y  represent  the  co-ordinates  of  any  point  P 
in  the  generating  curve,  and  M  the  volume  of  the  portion 
PAQ  of  this  solid,  then,  conceiving  it  to  be  made  up  of 
cylindrical  laminse  parallel  to  PQ,  the  thickness  of  each  of 
which  is  A#,  the  volume  of  each  is  represented  by  iry*&x,  and 
its  moment  by  *pxy*&x. 

«i 

*l  xy*dx 
^xy^x_^  ...... 

M  M 

EXAMPLE.  —  To  determine  the  centre  of  gravity  of  any  solid 
segment  of  a  sphere. 

Let  BjACj  represent  any  such  segment  of  a 
sphere  whose  centre  is  D  and  its  radius  a.  Let  x 
and  y  represent  the  co-ordinates  AM  and  MP  of 
any  point  P,  x  being  measured  from  A  ;  then  by 
the  equation  to  the  circle  y*=2ax—x*, 


:.  if  fxy*dx=«  fx  (2ax—x*)  dx—« 

a?2  o 

x,  x, 

Also,  M.=«fydx  =  *  f(2ax—x*)  dx= 


OF  THK  SEGMENT  OF  AN  AKCH.  35 

(31). 


a — x 

v  JL  A 

If  the  segment  become  a  hemisphere,  xt=a9  /.Gl=-|a. 
36.   The  centre  of  gravity  of  the  sector  of  a  circle. 

Let  CAB  represent  such  a  sector ;  conceive  the  arc  ADB 
to  be  a  polygon  of  an  infinite  number  of  sides 
and  lines,  to  be  drawn  from  all  the  angles  of  the 
polygon  to  the  centre  C  of  the  circle,  these  will 
divide  the  sector  into  as  many  triangles.  Now 
/B  the  centre  of  gravity  of  each  triangle  will  be  at 
a  distance  from  C  equal  to  f-  the  line  drawn  from  the  vertex 
C  of  that  triangle  to  the  bisection  of  its  base,  that  is  equal 
to  f  the  radius  of  the  circle,  so  that  the  centres  of  gravity  of 
all  the  triangles  will  lie  in  a  circular  arc  FE,  whose  centre  is 
C  and  its  radius  CF  equal  to  fCA,  and  the  weights  of  the 
triangles  may  be  supposed  to  be  collected  in  this  arc  FE, 
and  to  be  uniformly  distributed  through  it,  so  that  the  cen- 
tre of  gravity  G  of  the  whole  sector  CAB  is  the  centre  of 
gravity  of  the  circular  arc  FE.  Therefore  by  equation  (23), 
if  S1,  C1,  and  a\  represent  the  arc  FE,  its  chord  FE,  and  its 
radius  CF,  and  S,  C,  &,  the  similar  arc,  chord,  and  radius  of 

ADB,  then  CG  —       ^    ;  but  since  the  arcs  AB  and  FE  are 

similar,  and  that  a1  =  \a,  :.  C1  =  f  C  and  S1  =  f  S.     Substi- 
tuting these  values  in  the  last  equation,  we  have 

C*C*         2.  ^  /QO\ 

=|-s- (32)" 

37.  The  centre  of  gravity  of  any  portion  of  a  circular  ring 
or  of  an  arch  of  equal  voussoirs. 

2  represent  any  such  portion  of  a  circular  ring 
whose  centre  is  A.  Let  al  represent  the 
radius,  and  Ct  the  chord  of  the  arc  BA>  and 
Sj  its  length,  and  let  a»  C2  similarly  represent 
the  radius  and  chord  of  the  arc  B2C2,  and  Sa 
the  length  of  that  arc. 

Also  let  G,  represent  the  centre  of  gravity  of  the  sector 
~15  G2  that  of  the  sector  AB2C2,  and  Q-  the  centre  of 
gravity  of  the  ring.     Then 


AG2  x  sect,  AB2C2+  AG  x  ring  B1C1B2C2=  AG;  x  sect.  ABA 


ISTow  (by  equation  32),  AG1=-|i,    AG2=f 

8 


36  THE   PROPERTIES 

also  sector  AB1CI=  ^8^,  sector  AB2C2=  ^S2#2, 
.-.  ring  B1Ol01B1=sect.  AB.C,— sect.  AB2C2=  JSA 


.-.  AG  .  (SA— ^,0=1  (OA*—  CA*), 

.-.  AG  =  |  ^  ""  fy*«a (33). 

38.  THE  PROPERTIES  OF  GULDENUS. 

If  NL  represent  any  plane  area,  and  AB  be  any  axis,  in  the 
same  plane,  about  which  the  area  is  made  to 
revolve,  so  thai  NL  is  by  this  revolution  made  to 
generate  a  solid  of  revolution,  then  is  the  volume 
of  this  solid  equal  to  that  of  a  prism  whose  base 
^s  NL,  and  whose  height  is  equal  to  the  length 
of  the  path  which  the  centre  of  gravity  G  of  the 
area  NL  is  made  to  describe. 


For  take  any  rectangular  area  PRSQ  in  NL,  whose  sides 
are  respectively  parallel  and  perpendicular  to  AB,  and  let 
MT  be  the  mean  distance  of  the  points  P  and  Q,  or  R  and 
S,  from  AB.  Now  it  is  evident  that  in  the  revolution  of 
NL  about  AB,  PQ  will  describe  a  superficial  ring. 

Suppose  this  to  be  represented  by  QFPK,  let  M  be  the 
centre  of  the  ring,  and  let  the  arc  subtended  by 
the  angle  QMF  at  distance  unity  from  M  be  repre- 
sented  by  d,  then  the  area  FQPK  equals  the  sector 
FQM—  the  sector  ~ 


Now  the  solid  ring  generated  by  PRSQ  is  evidently  equal 
to  the  superficial  ring  generated  by  PQ,  multiplied  by  the 
distance  PR.  This  solid  ring  equals  therefore  6  (MT  x  PQ 
xPR)  or  dxMTxPRSQ.  Now  suppose  the  area  PRSQ 
to  be  exceedingly  small,  and  the  whole  area  NL  to  be  made 
up  of  such  exceedingly  small  areas,  and  let  them  be  repre- 
sented by  #15  a»  a3,  &c.  and  their  mean  distances  MT  by  x:, 
a?2,  a?3,  &c.  then  the  solid  annuli  generated  by  these  areas 
respectively  will  (as  we  have  shown),  be  represented  by 
&xzaz,  &c.  &c.  ;  and  the  sum  of  these  annuli, 


OF   GTJLDINTJS.  37 


or  the  whole  solid,  will  be  represented  by  .^^  ,  . 
to,a,  +  &c.,  or  by  6  (x^a, + x,a, + x,a3  +  &c.).  Now  if  p  repre- 
sent the  weight  of  any  superficial  el'ement  of  the  plane  NX, 
xla1^=thQ  moment  of  the  weight  of  a,  about  the  axis  AB* 
35,0^= that  of  the  area  <z2  about  the  same  axis  AB,  and  so 
on,  therefore  the  sum  (fl?1<^,-|-fiP,aj-hQ9taf-4i&o.)f*=;the  moment 
of  the  whole  area  NL  about  AB ;  but  if  G  be  the  centre  of 
gravity  of  NL,  and  GI  its  distance  from  AB,  then  the 
moment  of  NL  about  AB=GI> 
therefore  the  whole  solid =6  .  Gl 


but  6  .  GI  equals  the  length  of  the  circu- 
lar path  described  by  G  ;  therefore  the 
volume  of  the  solid  equals  NL  multi- 
plied by  the  length  of  the  path  de- 
scribed by  G,  i.  e.  it  equals  &^sm  NM, 
whose  base  is  NL,  and  whose  height  GH 
is  the  length  of  the  path  described  by 
G  ;  which  is  the  first  property  of  GUL- 
DDTCJS. 

39.  The  above  proposition   is   applicable  to  finding  the 
solid  contents  of  the  thread  of  a  screw  of  variable  diame- 
ter, or  of  the   material  in   a  spiral    staircase:    for    it  is 
evident  that  the  thread  of  a  screw  may  be  supposed  to  be 
made  up  of  an  infinite  number  of  small  solids  of  revolution, 
arranged  one  above  another  like  the  steps  of  a  staircase;  all 
of  which  (contained  in  one  turn  of  the  thread)  might  be 
made  to  slide  along  the  axis,  so  that  their  surfaces  should  all 
lie  in  the  same  plane  ;  in  which  case  they  would  manifestly 
form  one  solid  of  revolution,  such  as  that  whose  volume  has 
been  investigated.     The  principle  is  moreover  applicable  to 
determine  the  volume  of  any  solid  (however  irregular  may 
be  its  form  otherwise),  provided  only  that  it  may  be  con- 
ceived to  be  generated  by  the  motion  of  a  given  plane  area, 
perpendicular  to  a  given  curved  line,  which  always  passes 
through  the  same  point  in  the  plane.     For  it 
is  evident  that  whatever  point  in  this  curved 
line  the  plane  may  at  any  instant  be  traver- 
sing, it  may  at  that  instant  be  conceived  to  be  revolving 
about  a  certain  fixed  axis,  passing  through  the  centre  of 
curvature  of  the  curve  at  that  point;  and  thus  revolving 
about  a  fixed  axis,  it  is  generating  for  an  instant  a  solid  of 
revolution  about  that  axis,  the  volume  of  which  elementary 
solid  of  revolution  is  equal  to  the  area  of  the  plane  miilti- 


38  THE    PEOPEETIES 

plied  by  the  length  of  the  path  described  by  its  centre  of 
gravity ;  and  this  being  true  of  all  such  elementary  solids. 
each  being  equal  to  the'  product  of  the  plane  by  the  corres- 
ponding elementary  path  of  the  centre  of  gravity,  it  follows 
that  the  whole  volume  of  the  solid  is  equal  to  the  product 
of  the  area  by  the  whole  length  of  the  path. 

40.  If  AB  represent  any  curved  line  made  to  revolve  about 
the  axis  AD  so  as  to  generate  the  sur- 
face of  revolution  BAG,  and  G  l)e  the 
centre  of  gravity  of  this  curved  line, 
then  is  the  area  of  this  surface  equal 
to  the  product  of  the  length  of  the 
curved  line  AB,  by  the  length  of  the 
path  described  lyy  thepoint  G,  during 
the  revolution  of  the  curve  about  AD.     This  is  the  second 
property  of  Guldinus. 

Let  PQ  be  any  small  element  of  the  generating  curve, 
and  PQFK  a  zone  of  the  surface  generated  by  this  element, 
this  zone  may  be  considered  as  a  portion  of  the  surface  of  a 
cone  whose  apex  is  M,  where  the  tangents  to  the  curve  at  T 
and  V,  which  are  the  middle  points  of  PQ  and  FK,  meet 
when  produced.  Let  this  band  PQFK  of  the  cone  QMF  be 
developed*,  and  let  PQFK  represent  its  develop- 
ment ;  this  figure  PQFK  will  evidently  be  a  circu- 
lar ring,  whose  centre  is  M ;  since  the  develop- 
ment of  the  whole  cone  is  evidently  a  circular 
sector  MQF  whose  centre  M  corresponds  to  the 
apex  of  the  cone,  and  its  radius  MQ  to  the  side  MQ  of  the 
cone. 

Now,  as  was  shown  in  the  last  proposition,  the  area  of 
this  circular  ring  when  thus  developed,  and  therefore  of  the 
conical  band  before  it  was  developed,  is  represented  by 

6  .  MT  .  PQ,  where  0  represents  the  arc  subtended  by  QMF 
at  distance  unity.  Now  the  arc  whose  radius  is  MT  is 
represented  by  &  .  MT ;  but  this  arc,  before  it  wras  developed 
from  the  cone,  formed  a  complete  circle  whose  radius  was 
NT,  and  therefore  its  circumference  2^NT ;  since  then  the 
circle  has  not  altered  its  length  by  its  development,  we 
have 

*  If  the  cone  be  supposed  covered  with  a  flexible  sheet,  and  a  band  such 
as  PQFK  be  imagined  to  be  cut  upon  it,  and  then  unwrapped  from  the  cone 
and  laid  upon  a  plane,  it  is  called  the  development  of  the  band. 


OF   GULDINUS.        ^        J^lOTttT^^ 

<*OaUfr*ri*l 


Substituting  this  value  of  dMT  in  the  expression  for  the  area 
of  the  band  we  have 

area  of  zone  PQFK=2*  .  NT  .  PQ. 

Let  the  surface  be  conceived  to  be  divided  into  an  infinite 
number  of  such  elementary  bands,  and  let  the  lengths  of 
the  corresponding  elements  of  the  curve  AB  be  represented 
by  sl9  $„,  $3,  &c.  and  the  corresponding  values  of  NT  by  yl5 
2/2?  2/3?  &c-  Then  will  the  areas  of  the  corresponding  zones 
b 


e  represented  by  2tf?/1s1,  2tfy2s2,  2#y3s3,  &c.  and  the  area  of 
the  whole  surface  BAG  by  %*y1sl  +  2tfy2s2  +  2*y,«8  +  ....  or 
by  2-7r(y1s1  +  y2s2  +  y3«$3-f  ....).  But  since  G  is  the  centre  of 
gravity  of  the  curved  line  AB,  therefore  AB  .  GHjx  repre- 
sents the  moment  of  the  weight  of  a  uniform  thread  or  wire 
of  the  form  of  that  line  about  AD,  j*  being  the  weight  of 
each  unit  in  the  length  of  the  line  :  moreover,  this  moment 
equals  the  sum  of  the  moments  of  the  weights  s^,  s^,  $,«•, 
&c.  of  the  elements  of  the  line. 


/.AB  .  GH=y1«1 


__ 
Therefore     area     of     surface    BAC=2*AB     .     GH=AB 


But  2-rrGH  equals  the  length  of  the  circular  path  described 
by  G  in  its  revolution  about  AD.  Therefore,  &c. 

This  proposition,  like  the  last,  is  true  not  only  in  respect  to 
a  surface  of  revolution,  but  of  any  surface  generated  by  a 
plane  curve,  which  traverses  perpendicularly  another  curve 
of  any  form  whatever,  and  is  always  intersected  by  it  in  the 
same  point.  It  is  evident,  indeed,  that  the  same  demonstra- 
tion applies  to  both  propositions.  It  must,  however,  be  ob- 
served, that  neither  proposition  applies  unless  the  motion  of 
the  generating  plane  or  curve  be  such,  that  no  two  of  its  con- 
secutive positions  intersect  or  cross  one  another. 

41.  The  volume  of  any  truncated  prismatic  or  cylindrical 
lody  ABCD,  of  which  one  extremity  CD  is  perpendicular 
to  the  sides  of  the  prism,  and  the  other  AB  inclined  to 
them,  is  equal  to  that  of  an  upright  jwism  ABEF,  having 
for  its  lose  the  plane  AB,  and  for  its  height  the  perpen- 
dicular height  GN  of  the  centre  of  gravity  G  of  the  plane 
DC,  above  the  plane  of  AB. 

For  let  i  represent  the  inclination  of  the  plane  DC  to  AB  ; 


40  THE   PROPERTIES    OF   GULDINUS. 

take  m,  any  small  element  of  the  plane 
CD,  and  let  mr  be  a  prism  whose  base  is  m 
and  whose  sides  are  parallel  to  AD  and 
BC  ;  of  elementary  prisms  similar  to  which 
the  whole  solid  ABCD  may  be  supposed 
to  be  made  up.  Now  the  volume  of  this  prism,  whose  base 
is  m  and  its  height  mr,  equals  mr  xm  =  sec.  i  x  (mr  .  cos.  i) 
xm  =  sec.  «  x  (mr  .  sin.  mm)  m  =  sec.  *  x  mn  x  m. 

Therefore  the  whole  solid  equals  the  sum  of  all  such  pro- 
ducts as  mn  x  m,  each  such  product  being  multiplied  by  the 
constant  quantity  sec.  i,  or  it  is  equal  to  the  sum  just  spoken 
of,  that  sum  being  divided  by  cos.  i.  Let  this  sum  be  repre- 
sented by  2mn  x  m,  therefore  the  volume  of  the  solid  is  re- 

,          'Zmn  xm      AT   '  ^T.  , 

presented  by  —      JNow  suppose  CD  to  represent  a 

thin  lamina  of  uniform  thickness,  the  weight  of  each  square 
unit  of  which  is  f*,  then  will  the  weight  of  the  element  m  be 
represented  by  M*  X  m,  and  its  moment  about  the  plane  ABIsT 
by  M-  x  mn  x  m,  and  ^mn  x  m  will  represent  the  sum  of  the 
moments  of  all  the  elements  of  the  lamina  similar  to  m  about 
that  plane.     Now  by  Art.  15.  this  sum  equals  the  moment  of 
the  whole  weight  of  the  lamina  f*  x  CD  supposed  to  be  col- 
lected in  G,  about  that  plane.     Therefore 
I*  x  CD  x  ~NG=^mn  x  m, 
:.  T)D  x  NG  —  Zmn  x  m. 
Substituting  this  value  of  2mn  x  n,  we  have 

volume  of  solid  =  sec.  *  x  CD  x  NGL 

But  the  plane  CD  is  the  projection  of  AB,  therefore  CD 
=  AB  cos.  i,  .'.  CD  x  sec.  i  =  AB  ; 

.-.  vol.  of  solid  ABCD  =  AB  x  SX5  =  vol.  of  prism  ABEF. 
Therefore,  &c. 

[Q.  E.  D  .] 


MOTION.  41 


T    II. 

DYNAMICS. 

42.  MOTION  is  change  of  place. 

The  science  of  DYNAMICS  is  that  which  treats  of  the  laws 
which  govern  the  motions  of  material  bodies,  and  of  their 
relation  to  the  forces  whence  those  motions  result. 

The  SPACES  described  by  a  moving  body  are  the  distances 
between  the  positions  which  it  occupies  at  different  succes- 
sive periods  of  time. 

UNIFORM  MOTION  is  that  in  which  equal  spaces  are  de- 
scribed in  equal  successive  intervals  of  time. 

The  VELOCITY  of  uniform  motion  is  the  space  which  a 
body  moving  uniformly  describes  in  each  second  of  time. 
Thus  if  a  body  move  uniformly  with  a  velocity  represented 
by  Y,  and  during  a  time  represented  in  seconds  by  T,  then 
the  space  S  described  by  it  in  those  T  seconds  is  represented 

by  TY,  or  S=TY.  Whence  it  follows  that  Y  =  |-and  T=L5 

so  that  if  a  body  move  uniformly,  the  space  described  by  it 
is  equal  to  the  velocity  multiplied  by  the  time  in  seconds, 
the  velocity  is  equal  to  the  space  divided  by  the  time,  and 
the  time  is  equal  to  the  space  divided  by  the  velocity. 

43.  It  is  a  law  of  motion,  established  from  constant  obser- 
vation upon  the  motions  of  the  planets,  and  by  experiment 
upon  the  motions  of  the  bodies  around  us,  that  when  once 
communicated  to  a  body,  it  remains  in  that  body,  unaffected 
by  the  lapse  of  time,  carrying  it  forward  for  ever  with  the 
same  velocity  and  in  the  same  direction  in  which  it  first  be- 

n  to  move,  unless  some  force  act  afterwards  in  a  contrary 
irection  to  destroy  it.* 

*  This  is  the  first  LAW  OP  MOTION.  For  numerous  illustrations  of  this  fun- 
damental law  of  motion,  the  reader  is  referred  to  the  author's  work,  entitled, 
ILLUSTRATIONS  OF  MECHANICS,  Art.  193. 


42  VELOCITY. 

The  velocity,  at  any  instant,  of  a  body  moving  with  a 
VARIABLE  MOTION,  is  the  space  which  it  would  describe  in 
one  second  of  time  if  its  motion  were  from  that  instant  to 
become  UNIFORM. 

An  ACCELERATING  FORCE  is  that  which  acting  continually 
upon  a  body  in  the  direction  of  its  motion,  produces  in  it  a 
continually  increasing  velocity  of  motion. 

A  RETARDING  FORCE  is  that  which  acting  upon  a  body  in 
a  direction  opposite  to  that  of  its  motion  produces  in  it  a 
continually  diminishing  velocity. 

An  IMPULSIVE  FORCE  is  that  which  having  communicated 
motion  to  a  body,  ceases  to  act  upon  it  after  an  exceedingly 
small  time  from  the  commencement  of  the  motion. 


44.  A  UNIFORMLY  accelerating  or  retarding  force  is  that 
which  produces  equal  increments  or  decrements  of  velocity 
in  equal  successive  intervals  of  time.  If  f  represent  the 
additional  velocity  communicated  to  a  body  by  a  uniformly 
accelerating  force  in  each  successive  second  of  time,  and  T 
the  number  of  seconds  during  which  it  moves,  then  since  by 
the  first  law  of  motion  it  retains  all  these  increments  of  velo- 
city (if  its  motion  be  unopposed),  it  follows  that  after  T 
seconds,  an  additional  velocity  represented  by  f  T,  will  have 
been  communicated  to  it  ;  and  if  at  the  commencement  of 
this  T  seconds  its  velocity  in  the  same  direction  was  Y,  then 
this  initial  velocity  having  been  retained  (by  the  first  law  of 
motion),  its  whole  velocity  will  have  become  Y+/T. 

If,  on  the  contrary,  f  represents  the  velocity  continually 
taken  away  from  a  body  in  each  successive  second  of  time, 
by  a  uniformly  retarding  force,  and  Y  the  velocity  with 
which  it  began  to  move  in  a  direction  opposite  to  that  in 
which  this  retarding  force  acts,  then  will  its  remaining  velo- 
city after  T  seconds  be  represented  by  Y—  /T;  so  that  gene- 
rally the  velocity  Y  of  a  body  acted  upon  by  a  uniformly 
accelerating  or  retarding  force  is  represented,  after  T  seconds, 
by  the  formula 

(34). 


The  force  of  gravity  is,  in  respect  to  the  descent  of  bodies 
near  the  earth's  surface,  a  constantly  accelerating  force, 
increasing  the  velocity  of  their  descent  by  32  j  feet  in  each 
successive  second,  and  if  they  be  projected  upwards  it  is  a 
constantly  retarding  force,  diminishing  their  velocity  by  that 
quantity  in  each  second.  The  symbol  g  is  commonly  used  to 


VELOCITY.  43 

represent  this  number  32| ;  so  that  in  respect  to  gravity  the 
above  formula  becomes  v=V  ±0T,  the  sign  ±  being  taken 
according  as  the  body  is  projected  downwards  or  upwards. 

A  VARIABLE  accelerating  force  is  that  which  communicates 
unequal  increments  of  velocity  in  equal  successive  intervals 
of  time ;  and  a  variable  retarding  force  that  which  takes 
away  unequal  decrements  of  velocity.* 

45.    To   DETERMINE  THE  RELATION   BETWEEN   THE   VELOCITY  AND 
THE   SPACE,  AND   THE   SPACE   AND   TIME   OF   A   BODY'S   MOTION. 

Let  AM1?  M^,  M2M8,  &c.  represent  the  exceeding  small 
successive  periods  of  a  body's  motion,  and 
AP  the  velocity  with  which  it  began  to 
move,  MjPj  the  velocity  at  the  expiration 
of  the  first  interval  of  time,  M2P2  that  at 
the  expiration  of  the  second,  M3P3  of  the 
third  interval  of  time,  and  so  on;  and 
instead  of  the  body  varying  the  velocity  of  its  motion  con- 
tinually  throughout  the  period  AM1?  suppose  it  to  move 
through  that  interval  with  a  velocity  which  is  a  mean 
between  the  velocity  AP  at  A,  and  that  M^  at  M15  or  with 
a  velocity  equal  to  ^(AP  +  M.P,). 

Since  on  this  supposition  it  moves  with  a  uniform  motion, 
the  space  it  describes  during  the  period  AM1  equals  the 
product  of  that  velocity  by  that  period  of  time,  or  it  equals 
•KAP-fMjP^AMj.  Now  this  product  represents  the  area 
of  the  trapezoid  AM^P.  The  space  described  then  in  the 
interval  AMn  on  the  supposition  that  the  body  moves  during 
that  interval  with  a  velocity  which  is  the  mean  between 
those  actually  acquired  at  the  commencement  and  termi- 
nation of  the  interval,  is  represented  by  the  trapezoidal 
areaAM^P. 

Similarly  the  areas  P^,,  P2M3,  &c.  represent  the  spaces 
the  body  is  made  to  describe  in  the  successive  intervals 
MjM2,  M2M3,  &c. ;  and  therefore  the  whole  polygonal  area 
APCB  represents  the  whole  space  the  body  is  made  to 
describe  in  the  whole  time  AB,  on  the  supposition  that  it 
moves  in  each  successively  exceeding  small  interval  of  time 
with  the  mean  velocity  of  that  interval.  Now  the  less  the 
intervals  are,  the  more  nearly  does  this  mean  velocity  of  each 
interval  approach  the  actual  velocity  of  that  interval ;  and 
if  they  be  infinitely  small,  and  therefore  infinitely  great  in 

*  Kote  (i)  Ed.  App. 


Me  MOTION   UNIFORMLY 

number,  then  the  mean  velocity  coincides  with  the  actual 
velocity  of  each  interval,  and  in  this  case  the  polygonal  area 
passes  into  the  curvilinear  area  APCB. 

Generally,  therefore,  if  we  represent  by  the  abscissa  of  a 
curve  the  times  through  which  a  body  has  moved,  and  by 
the  corresponding  ordinates  of  that  curve  the  velocities  which 
it  has  acquired  after  those  times,  then  the  area  of  that  curve 
will  represent  the  space  through  which  the  body  has  moved ; 
or  in  other  words,  if  a  curve  PC  be  taken  such  that  the  num- 
ber of  equal  parts  in  any  one  of  its  abscissae  AM9  being  taken 
to  represent  the  number  of  seconds  during  which  a  body  has 
moved,  the  number  of  those  equal  parts  in  the  corresponding 
ordinate  M3P3  will  represent  the  number  of  feet  in  the  velo- 
city then  acquired;  then  the  space  which  the  body  has 
described  will  be  represented  by  the  number  of  these  equal 
parts  squared  which  are  contained  in  the  area  of  that  curve. 


46.  To  DETERMINE  THE  SPACE  DESCRIBED  IN  A  GIVEN  TIME  BY 
A  BODY  WHICH  IS  PROJECTED  WITH  A  GIVEN  VELOCITY,  AND 
WHOSE  MOTION  IS  UND7ORMLY  ACCELERATED,  OR  UNIFORMLY 
RETARDED. 

Take  any  straight  line  AB  to  represent  the  whole  time  T, 
in  seconds,  of  the  body's  motion,  and  draw  AD 
perpendicular  to  it,  representing  on  the  same 
scale  its  velocity  at  the  commencement  of  its 
motion.  Draw  DE  parallel  to  AB,  and  accord- 
ing as  the  motion  is  accelerated  or  retarded 
draw  DC  or  DF  inclined  to  DE,  at  an  angle  wThose  tangent 
equals/",  the  constant  increment  or  decrement  of  the  body's 
velocity.  Then  if  any  abscissa  AM  be  taken  to  represent  a 
number  of  seconds  t  during  which  the  body  has  moved,  the 
corresponding  ordinate  MP  or  MQ  will  represent  the  velocity 
then  acquired  by  it,  according  as  its  motion  is  accelerated  or 
retarded.  For  PR  =  EQ  =  DK  tan.  PDE= AM  tan.  PDE  ; 
but  AM  =  £,  and  tan.  PDE=/:  therefore  PR  =  KQ=/j{. 
Also  BM=AD=V,  therefore  JdP==BM+PB==V-f/^,  and 
MQ=EM— RQ=V- -ft\  therefore  by  equation  (34),  MP  or 
MQ  represents  the  velocity  after  the  time  AM  according  as 
the  motion  is  accelerated  or  retarded.  The  same  being  true 
of  every  other  time,  it  follows,  by  the  last  proposition,  that 
the  whole  space  described  in  the  time  T  or  AB  is  represented 
by  the  area  ABCD  if  the  motion  be  accelerated,  and  by  the 
area  ABFD  if  it  be  retarded. 


ACCELEEATED    OK   RETARDED.  45 

ISTow  area  ABCD=iAB(AD+BO),  but  AB=T,  AD=Y 
BC=Y+/T, 

/.  area  ABOD=iT(V+V+/T)=YT+i/T'. 

Also  area  ABFD=JAB  (AD  +  BF),  where  AB  and  AD 
have  the  same  values  as  before,  and  BF=Y— /T, 

/.  area  ABFD=iT(Y+Y-/T)^YT-i/T2. 

Therefore  generally,  if  S  represent  the  space  described  after 
T  seconds, 

S-YT±i/Ta (35); 

in  which  formula  the  sign  ±  is  to  be  taken  according  as  the 
motion  is  accelerated  or  retarded. 


47.  To  DETERMINE  A  RELATION  BETWEEN  THE  SPACE  DESCRD3ED 
AND  THE  VELOCITY  ACQUIRED  BY  A  BODY  WHICH  IS  PROJECTED 
WITH  A  GIVEN  VELOCITY,  AND  WHOSE  MOTION  IS  UNIFORMLY 
ACCELERATED  OR  RETARDED. 

Let  v  be  the  velocity  acquired  after  T  seconds,  then  by 
equation  (34),  v  =  Y  ±/T,  .-.  T  =  ±^—  • 

p  J*\ c        ]Sr°w  area  ABCI)  =  *AB  ( AD  +  BC)>  where 

E   AB=T=— — ^' 
p  f 


(v—  V} 
:.  area  ABCD=  i^1"  ("V+  ^)  = 


f 
area  ABFD  =  JAB  (AD+BF),  where  AB=T  =  - 


Therefore  generally,  if  S  represent  the  space  through  which 
the  velocity  v  is  acquired,  then  S=iJ-  —  3  —  4 

(36); 


in  which  formula  the  ±  sign  is  to  be  taken  according  as 
the  motion  is  accelerated  or  retarded. 

If  the  body's  motion  be  retarded,  its  velocity  v  will  eventu- 
ally be  destroyed.     Let  Sj  be  the  space  which  will  have  been 


4:6  THE   UNIT    OF   WORK. 

described  when  v  thus  vanishes,  then  by  the  last  equation 
0-Y2=  -  2/S,. 

A   Y2-2/S1 (37), 

where  Y  is  the  velocity  with  which  the  body  is  projected 
in  a  direction  opposite  to  the  force,  and  St  the  whole  space 
which  by  this  velocity  of  projection  it  can  be  made  to 
describe. 

If  the  body's  motion  be  accelerated,  and  it  fall  from  rest, 
or  have  no  velocity  of  projection,  then  ir2— -  0  =  +2/*S, 

.•.tf=2/S (38). 

Let  S2  be  the  space  through  which  it  must  in  this  case 
move  to  acquire  a  velocity  V  equal  to  that  with  which  it 
was  projected  in  the  last  case,  therefore  Y2=  2/*Sa.  Whence 
it  follows  that  S1=Sa,  or  that  the  whole  space  St  through 
which  a  body  will  move  when  projected  with  a  given  velo- 
city Y,  and  uniformly  retarded  by  any  force,  is  equal  to  the 
space  Sa,  through  which  it  must  move  to  acquire  that  velo- 
city when  uniformly  accelerated  by  the  same  force. 

In  the  case  of  bodies  moving  freely,  and  acted  upon  by 
gravity,  f  equals  32|  feet  and  is  represented  by  g ;  and  the 
space  Sa,  through  which  any  given  velocity  Y  is  acquired,  is 
then  said  to  be  that  due  to  that  velocity. 


WOBK. 

48.  WORK  is  the  union  of  a  continued  pressure  with  a 
continued  motion.     And  a  mechanical  agent  is  thus  said  to 
WORK  when  a  pressure  is  continually  overcome,  and  a  point 
(to  which  that  pressure  is  applied)  continually  moved  by  it. 
Neither  pressure  nor  motion  alone  is  sufficient  to  constitute 
work /  so  that  a  man  who  merely  supports  a  load  upon  his 
shoulders,  without  moving  it,  no  more  works,  in  the  sense  in 
which  that  term  is  here  used,  than  does  a  column  which  sus- 
tains a  heavy  weight  upon  its  summit ;  and  a  stone,  as  it  falls 
freely  in  vacuo,  no  more  works  than  do  the  planets  as  they 
wheel  unresisted  through  space.* 

49.  THE  UNIT  OF  WORK. — The  unit  of  work  used  in  this 
country,  in  terms  of  which  to  estimate  every  other  amount 

*  Note  (,;• )  Ed.  App. 


VARIABLE   WORK.  47 

of  work,  is  the  work  necessary  to  overcome  a  pressure  of  one 
pound  through  a  distance  of  one  foot,  in  a  direction  opposite 
to  that  in  which  a  pressure  acts.  Thus,  for  instance,  if  a 
pound  weight  be  raised  through  a  vertical  height  of  one  foot, 
one  unit  of  work  is  done ;  for  a  pressure  of  one  pound  is 
overcome  through  a  distance  of  one  foot,  in  a  direction  oppo- 
site to  that  in  which  the  pressure  acts. 

50.  The  number  of  units  of  work  necessary  to  overcome  a 
pressure  of  M  pounds  through  a  distance  of  1ST  feet,  is 
equal  to  the  product  MN. 

For  since,  to  overcome  a  pressure  of  one  pound  through 
one  foot  requires  one  unit  of  work,  it  is  evident  that  to  over- 
come a  pressure  of  M  pounds  through  tho  same  distance  of 
one  foot,  will  require  M  units.  Since,  then,  M  units  of  work 
are  required  to  overcome  this  pressure  through  one  foot,  it 
it  evident  that  N"  times  as  many  units  (i.  e.  KM)  are  required 
to  overcome  it  through  IN"  feet.  Thus,  if  we  take  U  to  repre- 
sent the  number  of  units  of  work  done  in  overcoming  a  con- 
stant pressure  of  M  pounds  through  N  feet,  we  have 

(39).* 


51.    To    ESTIMATE    THE    WOEK    DONE    UNDER    A    VARIABLE 
PRESSURE. 

Let  PC  be  a  curved  line  and  AB  its  axis,  such  that  any 
one  of  its  abscissae  AM3,  containing  as  many 
equal  parts  as  there  are  units  in  the  space 
through  which  any  portion  of  the  work  has 
M¥  M  been  done,  the  corresponding  ordinate  MSP3 

may  contain  as  many  of  those  equal  parts, 
as  there  are  in  the  pressure  under  which  it  is  then  being 
done.  Divide  AB  into  exceedingly  small  equal  parts,  AM19 
MjM2,  &c.,  and  draw  the  ordinates  MjPj,  M2P2,  &c. ;  then  if 
we  conceive  the  work  done  through  the  space  AMj  (which 
is  in  reality  done  under  pressures  varying  from  AP  to  M,?,), 
to  be  done  uniformly  under  a  pressure,  which  is  the  arith- 
metic mean  between  AP  and  M^,  it  is  evident  that  the 
number  of  units  in  the  work  done  through  that  small  space 
will  equal  the  number  of  square  units  in  the  trapezoid 
APPjM,  (see  Art.  45.),  and  similarly  with  the  other  trape- 

*  Note  (fc)  Ed.  App. 


48  THE   RESOLUTION 

zoids  ;  so  that  the  number  of  units  in  the  whole  work  done 
through  the  space  AB  will  equal  the  number  of  square  units 
in  the  whole  polygonal  area  APPff^  &c.,  CB. 

But  since  AM,,  M.M^  &c.,  are  exceedingly  small,  'this 
polygonal  area  passes  into  the  curvilinear  area  APCB  ;  the 
whole  work  done  is  therefore  represented  by  the  number  of 
square  equal  parts  in  this  area. 

Now,  generally,  the  area  of  any  curve  is  represented  by 

the  integral  I  ydx,  where  y  represents  the  ordinate,  and  x 

the  corresponding  abscissa.  But  in  this  case  the  variable 
pressure  P  is  represented  by  the  ordinate,  and  the  space  S 
described  under  this  variable  pressure  by  the  abscissa.  If 
therefore  U  represent  the  work  done  between  the  values  Sa 
and  Sa  of  S,  we  have. 

s 


(40). 

8, 

Mean  pressure  is  that  under  which  the  same  work  would 
be  done  over  the  same  space,  provided  that  pressure,  instead 
of  varying  throughout  that  space,  remained 
the  same  :  thus,  the  mean  pressure  in  re- 
spect to  an  amount  of  work  represented  by 
the  curvilinear  area  AEFC,  is  that  under 
which  an  amount  of  work  would  be  done 
represented  by  the  rectilineal  area  ABDC,  the  area  ABDC 
being  equal  to  the  curvilinear  area  AEFC  ;  the  mean  pres- 
sure in  this  case  is  represented  by  AB.  Thus,  to  determine 
the  mean  pressure  in  any  case  of  variable  pressure,  we  have 
only  to  find  a  curvilinear  area  representing  the  work  done 
under  that  variable  pressure,  and  then  to  describe  a  rectan- 
.gular  parallelogram  on  the  same  base  AC,  which  shall  have 
an  area  equal  to  the  curvilinear  area. 

If  S  represent  the  space  described  under  a  variable  pres- 
sure, U  the  work   done,  and  p  the  mean  pressure,  then 

pS  =  U,  therefore  p  =  -^  .* 

52.  To  estimate  the  work  of  a  pressure,  whose  direction  is  not 
that  in  which  its  point  of  application  is  made  to  move. 

Hitherto  the  work  of  a  force  has  been  estimated  only  on 

*  Note  (I)  Ed.  App. 


OF   WOBK. 


the  supposition  that  the  point  of  applica- 
„-  tion  of  that  force  is  moved  in  the  direction 

in  which  the  force  operates,  or  in  the  oppo- 


site direction.  Let  PQ  be  tfye  direction  of 
a  pressure,  whose  point  of  application  Q 
is  made  to  move  in  the  direction  of  the 
straight  line  AB.  Suppose  the  pressure  P  to  remain  con- 
stant, and  its  direction  to  continue  parallel  to  itself.  It  is 
required  to  estimate  the  work  done,  whilst  the  point  of 
application  has  been  moved  from  A  to  Q. 

Eesolve  P  into  R  and  S,  of  which  R  is  parallel  and  S  per- 
pendicular to  AB.  Then  since  no  motion  takes  place  in  the: 
direction  of  SQ,  the  pressure  S  does  no  work,  and  the  whole.- 
work  is  done  by  R ;  therefore  the  work  =  R  .  AQ. 

]STow  R=P  .  cos.  PQR,  therefore  the  work=:P  .  AQ»cos;. 
PQR.  From  the  point  A  draw  AM  perpendicular  to  PQ> 
then  AQ  cos.  PQR^QM ;  therefore  work^P  .  QM.  There- 
fore the  work  of  any  pressure  as  above,  not  acting  ia  the 
direction  of  the  motion  of  the  point  of  application  of  that 
pressure,  is  the  same  as  it  would  have  been  if  the  point  of 
application  had  been  made  to  move  in  the  direction  of  the 
pressure,  provided  that  the  space  through  which  it  was  so 
moved  had  been  the  projection  of  the  space  through  which 

it  actually  moves.     The  product  P  .  QM  may  be  called  the 
work  of  r  resolved  in  the  direction  of  P. 

The  above  proposition  which  has  been  proved,  whatever 
may  be  the  distance  through  which  the  point  of  application 
is  moved,  in  that  particular  case  only  in  which  the  pressure 
remains  the  same  in  amount  and  always  parallel  to  itself,  is 
evidently  true  for  exceedingly  small  spaces  of  motion,  even 
if  the  pressure  be  variable  both  in  amount  and  direction ; 
since  for  such  exceedingly  small  variations  in  the  positions 
of  the  points  of  application,  the  variations  of  the  pressures 
themselves,  both  in  amount  and  direction,  arising  from  these 
variations  of  position,  must  be  exceedingly  small,  and  there- 
fore the  resulting  variations  in  the  work  exceedingly  small 
as  compared  with  the  whole  work.* 

*  Note  (m)  Ed.  App. 


50  THE   WOKK   OF 


53.  If  any  number  of  pressures  P,,  P2,  P3,  le  applied  to  the 
same  point  A,  and  remain  constant  and  parallel  to  them- 
selves, whilst  the  point  A  is  made  to  move  through  the 
straight  lirfe  AB,  then  the  whole  work  done  is  equal  to  the 
sum  o£  the  works  of  the  different  pressures  resolved  in  the 
directions  of  those  pressures,  each  being  taken  negatively 
whose  point  of  application  is  made  to  move  in  an  opposite 
to  the  pressure  upon  it. 


Let  ai?  a25  ag?  &c.  represent  the  inclinations  of  the  pres- 
sures P,,  P2,  &c.  to  the  line  AB,  then  will 
the  resolved  parts  of  these  pressures  in  the 
direction  of  that  line  be  Yl  cos.  al?  P2  cos. 
«2,  P3  cos.  «3,  &c.  and  they  will  be  equiva- 
lent to  a  single  pressure  in  the  direction 
of  that  line  represented  by  P,  cos.  «t  -f  P2 
cos.  a2  +  P3  cos.  a35  &c.  in  which  sum  all 
those  terms  are  to  be  taken  negatively  which  involve  pres- 
sures whose  direction  is  from  B  towards  A  (since  the  single 
pressure  from  A  towards  B  is  manifestly  equal  to  the  differ- 
ence between  the  sum  of  those  resolved  pressures  which  act 
in  that  direction,  and  those  in  the  opposite  direction).  There- 
fore the  whole  work  is  equal  to  jPj  cos.  ^  +  P2  cos.  «2  -f-  P3 
cosv  .....  }.  AB  =  P,  .  AB  cos,  a,  +  P2  .  AB  cos.  a2 


+  P3ABcos.  a3  +   ...    =P1 

;   in  which  expression  the  successive  terms  are  the 

works  of  the  different  pressures  resolved  in  the  several 
directions  of  those  pressures,  each  being  taken  positively  or 
negatively,  according  as  the  direction  of  the  corresponding 
pressure  is  towards  the  direction  of  the  motion  or  opposite 
to  it. 

Thus  if  U  represent  the  whole  work  and  Uj  and  U2  the 
sums  of  those  done  in  opposite  directions,  then 
U=U,-U,« (41). 

54.  If  any  number  of  pressures  applied  to  a  point  he  in  equi- 
librium, and  their  point  of  application  he  moved,  the  whole 
work  done  by  these  pressures  in  the  direction  of  the  motion 
•  will  equal  the  whole  work  done  in  the  opposite  direction. 

For  if  the  pressures  P,,  P2,  P3,  &c,  (Art.  53)  be  in  equi- 
librium, then  the  sums  of  their  resolved  pressures  in  opposite 

*  Note  (n)  Ed.  App. 


CENTRAL   FORCES.  51 

directions  along  AB  will  be  equal  (Art.  10) ;  therefore  the 
whole  work  U  along  AB,  which  by  the  last  proposition  is 
equal  to  the  work  of  a  pressure  represented  by  the  difference 
of  these  sums,  will  equal  nothing,  therefore  0  =  tJ,— U2, 
therefore  IT,— U2,  that  is,  the  whole  work  done  in  one  direc- 
tion along  AB,  by  the  pressures  P,,  P2,  &c.  is  equal  to  the 
whole  work  done  in  the  opposite  direction. 

55.  If  a  body  be  acted  upon  by  a  force  whose  direction  is 
always  towards  a  certain  point  S,  called  a  centre  of  force, 
and  be  made  to  describe  any  given  curve  PA  in  a  direction 
opposed  to  the  action  of  that  force,  and  Sp  be  measured  on 
SA  equal  to  SP,  then  will  the  work  done  in  moving  the 
body  through  the  curve  PA  be  equal  to  that  which  would 
be  necessary  to  move  it  in  a  straight  line  from  p  to  A. 

For  suppose  the  curve  PA  to  be  a  portion  of  a  polygon  of 
an  infinite  number  of  sides,  PP,,  P,P2,  &c. 
Through  the  points  P,  P,,  P2,  &c.  describe  circu- 
lar arcs  with  the  radii  SP,  SP,,  SP2,  &c.  and  let 
them  intersect  S  A  in  p,  p^  p^  &c.  Then  since 
PP,  is  exceedingly  small,  the  force  may  be  consi- 
dered to  act  throughout  this  space  always  in  a 
direction  parallel  to  SP, ;  therefore  the  work  done 
through  PP,  is  equal  to  the  work  which  must  be 
done  to  move  the  body  through  the  distance  raP,  (Art.  52.), 
since  mP,  is  the  projection  of  PP,  upon  the  direction  SP,  of 
the  force.  But  mPl=ppl ;  therefore  the  work  done  through 
PP,  is  equal  to  that  which  would  be  required  to  move  the 
body  along  the  line  S  A  through  the  distance  pp^ ;  and  simi- 
larly the  work  done  through  P,P2  is  equal  to  that  which 
must  be  done  to  move  the  body  through  p^p»  so  that 
the  work  through  PP2  is  equal  to  that  through  pp^  and  so 
of  all  other  points  in  the  curve.  Therefore  the  work  through 
PA  is  equal  to  that  through  pA.*  Therefore,  &c.  [Q.E.D.] 


*  Of  course  it  is  in  this  proposition  supposed  that  the  force,  if  it  be  not 
constant,  is  dependant  for  its  amount  only  on  the  distance  of  the  point  at 
which  it  acts  from  the  centre  of  force  S ;  so  that  the  distances  of  p  and  P 
from  S  being  the  same,  the  force  at  p  is  equal  to  that  at  P ;  similarly  the 
force  at  pi  is  equal  to  that  at  P1}  the  force  at  p*  equal  to  that  at  P2,  &c. 


52  THE    WOBK    OF 

56.  If  S  ~be  at  an  exceedingly  great  distance  as  compared 
with  AJ?,  then  all  the  lines  drawn  from  S  to  AP  may  ~be  con- 
sidered parallel.  This  is  the  case  with  the  force  of  gravity 
at  the  surface  of  the  earth,  which  tends  towards  a  point,  the 
earth's  centre,  situated  at  an  exceedingly  great  distance,  as 
compared  with  any  of  the  distances  through  which  the  work 
of  mechanical  agents  is  usually  estimated. 

Thus  then  it  follows  that  the  work  necessary  to  move  a 
heavy  body  up  any  curve  PA,  or  inclined  plane,  is  the  same 
as  would  be  necessary  to  raise  it  in  a  vertical  line  pA.  to  the 
same  height. 

The  dimensions  of  the  body  are  here  supposed  to  be  ex- 
ceeding small.  If  it  be  of  considerable  dimensions,  then 
whatever  be  the  height  through  which  its  centre  of  gravity 
is  raised  along  the  curve,  the  work  expended  is  the  same 
(Art  60.)  as  though  the  centre  of  gravity  were  raised  verti- 
cally to  that  height.* 


57.  In  the  preceding^  propositions  the  work  has  been  esti- 
mated on  the  supposition  that  the  body  is  made  to  move  so 
as  to  increase  its  distance  from  the  centre  S,  or  in  a  direction 
opposed  to  that  of  the  force  impelling  it  towards  S.  It  is 
evident,  nevertheless  that  the  work  would  have  been  precisely 
the  same,  if  instead  of  the  body  moving  from  P  to  A  it  had 
moved  from  A  to  P,  provided  only  that  in  this  last  case 
there  were  applied  to  it  at  every  point  such  a  force  as  would 
prevent  its  motion  from  being  accelerated  by  the  force  con- 
tinually impelling  it  towards  S ;  for  it  is  evident  that  to  pre- 
vent this  acceleration,  there  must  continually  be  applied  to 
the  body  a  force  in  a  direction  from  S  equal  to  that  by  which 
it  is  attracted  towards  it ;  and  the  work  of  such  a  force  is 
manifestly  the  same,  provided  the  path  be  the  same,  whether 
the  body  move  in  one  direction  or  the  other  along  that  path, 
being  in  the  two  cases  the  work  of  the  same  force  over  the 
same  space,  but  in  opposite  directions. 

*  The  only  force  acting  upon  the  body  is  in  this  proposition  supposed  to  be 
that  acting  towards  S.  No  account  is  taken  of  friction  or  any  other  forces 
"which  oppose  themselves  to  its  motion. 


PARALLEL   FORCES.  53 


58.  If  there  ~be  any  number  of  parallel  pressures,  Pn  PaJ  P3, 
&c.  whose  points  of  application  are  transferred,  each 
through  any  given  distance  from  one  position  to  another, 
then  is  the  work  which  would  he  necessary  to  transfer  their 
resultant  through  a  space  equal  to  that  ~by  which  their 
centre  of  pressure  is  displaced  in  this  change  of  position, 
equal  to  the  difference  between  the  aggregate  work  of  those 
pressures  whose  points  of  application  have  been  moved  in 
the  directions  in  which  the  pressures  applied  to  them  act, 
and  those  whose  points  of  application  have  been  moved  in 
the  opposite  directions  to  their  pressures. 

For  (Art.  IT.),  if  y15  ya,  ?/3,  &c.  represent  the  distances  of 
the  points  of  application  of  these  pressures  from  any  given 
plane  in  their  first  position,  and  h  the  distance  of  their  centre 
of  pressure  from  that  plane,  and  if  Yt,  Y2,  Y8  &c.  and  H  re- 
present the  corresponding  distances  in  the  second  position, 
and  if  P15  P2,  P3,  &c.  be  taken  positively  or  negatively  ac- 
cording as  their  directions  are  from  or  towards  the  given 
plane,  h  { 


,  (Y3-y2) 

'+P3(Y3-y3)  +  .....  (42); 

in  the  second  member  of  which  equation  the  several  terms 
are  evidently  positive  or  negative,  according  as  the  pressure 
P  corresponding  to  each,  arid  the  difference  Y—  y  of  its  dis- 
tances from  the  plane  in  its  two  positions,  have  the  same  or 
contrary  signs.  Now  by  supposition  P  is  positive  or  negative 
according  as  it  acts  from  or  towards  the  plane  ;  also  Y—  y  is 
evidently  positive  or  negative  according  as  the  point  of  appli- 
cation of  P  is  moved  from  or  towards  the  plane  ;  each  term 
is  therefore  positive  or  negative,  according  as  the  correspond- 
ing point  of  application  is  transferred  in  a  direction  towards 
that  in  which  its  applied  pressure  acts,  or  in  the  opposite 
direction. 

Now  the  plane  from  which  the  distances  of  the  points  of 
application  are  measured  may  be  any  plane  whatever.  Let 
it  be  a  plane  perpendicular  to  the  directions  of  the  pressures. 


54  THE   WORK   OF 


Let  A.xy  represent  this  plane,  and  let  P 
P'  represent  the  two  positions  of  the  point 
of  application  of  the  pressure  P  (the  path 
described  by  it  between  these  two  positions 
having  been  any  whatever).  Let  MP  and 
M'P'  represent  the  perpendicular  dis- 
tances of  the  points  P  and  P'  from  the 
plane,  and  draw  Pm  from  P  perpendicular 


to  M'P'.  Then  P  (Y— y)=P(M'P'-MP)^P  .  mP';  but  by 
Art.  55.,  P  .  mP'  equals  the  work  of  P  as  its  point  of  applica- 
tion is  transferred  from  P  to  P'.  Thus  each  term  of  the  second 
member  of  equation  (42)  represents  the  work  of  the  corre- 
sponding pressure,  so  that  if  2-z^,  represent  the  aggregate 
work  of  those  pressures  whose  points  of  application  are  trans- 
ferred towards  the  directions  in  which  the  pressures  act,  and 
2^2  the  work  of  those  whose  points  of  application  are  moved 
opposite  to  the  directions  in  which  they  severally  act,  then 
the  second  member  of  the  equation  is  represented  by  2^— 
2-z/,a.  Moreover  the  first  member  of  the  equation  is  evidently 
the  work  necessary  to  transfer  the  resultant  pressure  P2  + 
P2-f  P3  &c.  through  the  distance  H— A,  which  is  that  by 
which  the  centre  of  pressure  is  removed  from  or  towards  the 
given  plane,  so  that  if  U  represent  the  quantity  of  work 
necessary  to  make  this  transfer  of  the  centre  of  pressure, 

11=2^-2^ (43). 

t 

59.  If  the  sum  of  those  parallel  pressures  whose  tendency 
is  in  one  direction  equal  the  sum  of  those  whose  tendency 

is  in  the  opposite  direction,  then  Pj  +  Pg  +  Pg-j- =0. 

In  this  case,  therefore,  11=0,  therefore  2^—2^—0,  there- 
fore 2^=  2^2 ;  so  that  when  in  any  system  of  parallel  pres- 
sures the  sum  of  those  whose  tendency  is  in  one  direction 
equals  the  sum  of  those  whose  tendency  is  in  the  opposite  direc- 
tion, then  the  aggregate  work  of  those  whose  points  of  appli- 
cation are  moved  in  the  directions  of  the  pressure  severally 
applied  to  them  is  equal  to  the  aggregate  work  of  those  whose 
points  of  application  are  moved  in  the  opposite  directions. 

This  case  manifestly  obtains  when  the  parallel  pressures 
are  in  EQUILIBRIUM,  the  sum  of  those  whose  tendency  is  in 
one  direction  then  equalling  the  sum  of  those  whose  tendency 
is  in  the  opposite  direction,  since  otherwise,  when  applied  to 
a  point,  these  pressures  could  not  be  in  equilibrium  about 
that  point  (Art.  8.). 


PAEALLEL   FORCES.  55 

60.  The  preceding  proposition  is  manifestly  true  in  respect 
to  a  system  of  weights,  these  being  pressures  whose  directions 
are  always  parallel,  wherever  their  points  of  application  may 
be  moved.  'Now  the  centre  of  pressure  of  a  system  of 
weights  is  its  centre  of  gravity  (Art.  19).  Thus  then  it  fol- 
lows, that  if  the  weights  composing  such  a  system  be  sepa- 
rately moved  in  any  directions  whatever,  and  through  any 
distances  whatever,  then  the  difference  between  the  aggre- 
gate work  done  upwards  in  making  this  change  of  relative 
position  and  that  done  downwards  is  equal  to  the  work 
necessary  to  raise  the  sum  of  all  the  weights  through  a  height 
equal  to  that  through  which  their  centre  of  gravity  is  raised 
or  depressed.*  Moreover  that  if  .such  a  system  of  weights 
be  supported  in  equilibrium  by  the  resistance  of  any  fixed 
point  or  points,  and  be  put  in  motion,  ttVn  (since  the  work 
of  the  resistance  of  each  such  point  is  nothing)  the  aggregate 

*  This  proposition  has  numerous  applications.  If,  for  instance,  it  be  required 
to  determine  the  aggregate  expenditure  of  work  in  raising  the  different  ele- 
ments of  a  structure,  its  stone,  cement,  &c.,  to  the  different  positions  they 
occupy  in  it,  we  make  this  calculation  by  determining  the  work  requisite  to 
raise  the  whole  weight  of  material  at  once^to  the  height  of  the  centre  of  gra- 
vity of  the  structure.  If  these  materials  have  been  carried  up  by  labourers,  and 
we  are  desirous  to  include  the  whole  of  their  labour  in  the  calculation,  we 
ascertain  the  probable  amount  of  each  load,  and  conceive  the  weight  of  a  la- 
bourer to  be  added  to  each  load,  and  then  all  these  at  once  to  be  raised  to  the 
height  of  the  centre  of  gravity. 

Ag:iin,  if  it  be  required  to  determine  the  expenditure  of  work  made  in  rais- 
ing the  material  excavated  from  a  well,  or  in  pumping  the  water  out  of  it,  we 
know  that  (neglecting  the  effect  of  friction,  and  the  weight  and  rigidity  of  the 
cord)  this  expenditure  of  work  is  the  same  as  though  the  whole  material  had 
been  raised  at  one  lift  from  the  centre  of  gravity  of  the  shaft  to  the  surface. 
Let  us  take  another  application  of  this  principle  which  offers  so  many  practical 
results.  The  material  of  a  railway  excavation  of  considerable  length  is  to  be 
removed  so  as  to  form  an  embankment  across  a  valley  at  some  distance,  and  it 
is  required  to  determine  the  expenditure  of  work  made  in  this  transfer  of  th3 
material.  Here  each  load  of  material  is  made  to  traverse  a  different  distance, 
a  resistance  from  the  friction,  &c.,  of  the  road  being  continually  opposed  to  its 
motion.  These  resistances  on  the  different  loads  constitute  a  system  of  paral- 
lel pressures,  each  of  whose  points  of  application  is  separately  transferred  fro  n 
one  given  point  to  another  given  point,  tb.e  directions  of  transfer  being  als  > 
parallel.  Now  by  the  preceding  proposition,  the  expenditure  of  work  in  all 
these  separate  transfers  is  the  same  as  it  would  have  been  had  a  pressure  equil 
to  the  sum  of  all  these  pressures  been  at  once  transferred  from  the  centre  of 
resistance  of  the  excavation  to  the  centre  of  resistance  of  the  embankment. 
Now  the  resistances  of  the  parts  of  the  mass  moved  are  the  frictions  of  its  ele- 
ments upon  the  road,  and  these  frictions  are  proportional  to  the  weights  of  the 
elements  ;  their  centre  of  resistance  coincides  therefore  with  the  centre  of  gra- 
vity of  the  mass,  and  it  follows  that  the  expenditure  of  work  is  the  same  as 
though  all  the  material  had  been  moved  at  once  from  the  centre  of  gravity  of 
the  excavation  to  that  of  the  embankment.  To  allow  for  the  weight  of  the 
carriages,  as  many  times  the  weight  of  a  carriage  must  be  added  to  the  weight 
of  the  material  as  there  are  journeys  made. 


56  STABILITY   OF   THE   CENTRE. 

work  of  those  weights  which  are  made  to  descend,  is  equal 
to  that  of  those  which  are  made  to  ascend. 

61.  If  a  plane  ~be  taken  perpendicular  to  the  directions  of  any 
number  of  parallel  pressures  and  there  he  two  different  po- 
sitions ojr  the  points  of  application  of  certain  of  these  pi^es- 
sures  in  which  they  are  at  different  distances  from  the 
plane-,  whilst  the  points  of  application  of  the  rest  of  tJwse 
pressures  remain  at  the  same  distance  from  that  plane, 
and  if  in  both  positions  the  system  be  in  equilibrium,  then 
the  centre  of  pressure  of  the  first  mentioned  pressures  will 
be  at  the  same  distance  from  the  plane  in  both  positions. 

For  since  in  both  positions  the  system  is  in  equilibrium, 
therefore  in  both  positions  Pj  +  P2  +  P8  +   ...  =0, 


Now  let  Pw  be  any  one  of  the  pressures  whose  points  ol  appli- 
cation is  at  the  same  distance  from  the  given  plane  in  both 
positions, 

•••  Yn=yn,  and  Y.-y.  =  0, 
.•.(Y1-2/1)P1  +  (Y2- 
.-.  Y^+YJP,  +.  .  . 
.  Y1P1+Y2P2+  .  .  . 


where  Hw_,  represents  the  distance  of  the  centre  of  pressure 
of  P15  P2  .  .  .  Pn_,,  from  the  given  plane  in  the  first  position, 
and  An_1  its  distance  in  the  second  position.  Its  distance  in 
the  first  position  is  therefore  the  same  as  in  the  second, 
Therefore,  &c. 

From  this  proposition,  it  follows  that  if  a  system  of  weights 
be  supported  by  the  resistances  of  one  or  more  fixed  points, 
and  if  there  be  any  two  positions  whatever  of  the  weights  in 
both  of  which  they  are  in  equilibrium  with  the  resistances 
of  those  points,  then  the  height  of  the  common  centre  of 
gravity  of  the  weights  is  the  same  in  both  positions.  And 
that  if  there  be  a  series  of  positions  in  all  of  which  the 
weights  are  in  equilibrium  about  such  a  resisting  point  or 
points,  then  the  centre  of  gravity  remains  continually  at  the 
same  height  as  the  system  passes  through  this  series  of  posi- 
tions. 

If  all  these  positions  of  equilibrium  be  infinitely  near  to 


WORK   OF   PRESSURES.  57 

one  another,  then  it  is  only  during  an  infinitely  small  motion 
of  the  points  of  application  that  the  centre  of  gravity  ceases 
to  ascend  or  descend ;  and,  conversely,  if  for  an  infinitely 
small  motion  of  the  points  of  application  the  centre  of 
gravity  ceases  to  ascend  or  descend,  then  in  two  or  more 
positions  of  the  points  of  application  of  the  system,  infi- 
nitely near  to  one  another,  it  is  in  equilibrium. 

WORK  OF  PRESSURES  APPLIED  IN  DIFFERENT  DIRECTIONS  TO 
A  BODY  MOVEABLE  ABOUT  A  FIXED  AxiS. 

62.  The  work  of  a  pressure  applied  to  a  body  moveable  about 
a  fixed  axis  is  the  same  at  whatever  point  in  its  proper 
direction  that  pressure  may  be  applied. 

For  let  AB  represent  the  direction  of  a  pressure  applied 
to  a  body  moveable  about  a  fixed  axis 
O ;  the  work  done  by  this  pressure 
will  be  the  same  whether  it  be  ap- 
plied at  A  or  B.  For  conceive  the 
body  to  revolve  about  O,  through  an 
exceedingly  small  angle  AOC,  or 
BOD,  so  that  the  points  A  and  D  may  describe  circular  arcs 
AC  and  BD.  Draw  Cm,  Dn,  and  OE,  perpendiculars  to 
AB,  then  if  P  represent  the  pressure  applied  to  AB,  P  .  Am 
will  represent  the  work  done  by  P  when  applied  at  A  (Art. 
52.),  and  P  .  l&n  will  represent  the  work  done  by  P  when 
applied  at  B ;  therefore  the  work  done  by  P  at  A  is  the  same 
as  that  done  by  P  at  B,  if  Am  is  equal  to  B^. 

Now  AC  and  BD  being  exceedingly  small,  they  may  be 
conceived  to  be  straight  lines.  Since  BD  and  BE  are 
respectively  perpendicular  to  OB  and  OE,  therefore  /DBE 
—  /  BOE  ;'*  and  because  AC  and  AE  are  perpendicular  to 
O  A  and  OE,  therefore  /  CAE  =  /_  AOE.  Now  Am  = 

CA 
CA  .  cos.  CAE  =  CA  .  cos.  AOE  =  ^  .  OA  .  cos.  AOE 


PA 

=  7.  -  x  OE.     Similarly  B^  =  DB  cos.  DBE  =  DB  .  cos. 


BOE  =  55       OB  cos.  BOE  =    ^  x  OE,  i.e.  Am  =  OE  . 


*  It  is  a  well-known  principle  of  Geometry,  that  if  two  lines  be  inclined  at 
any  angle,  and  any  two  others  be  drawn  perpendicular  to  these,  then  the  incli- 
nation of  the  last  two  to  one  another  shall  equal  that  of  the  first  two. 


58  THE   ACCUMULATION   OF   WORK. 

CA  nBD  CA     BD 

PT-T-,  and  tin=  OE  -^5.    But  7—=7^^  since  the  /_  AOC— 

V/JTX  \)JL>  W-OL         \J±J 

/  BOD,  therefore  Am  =  ~Bn* 


63.  If  any  number  of  pressures  oe  ^n  equilibrium  about  a 
fixed  axis,  then  the  whole  work  of  those  which  tend  to  move 
the  system  in  one  direction  about  that  axis  is  equal  to  the 
whole  work  of  those  which  tend  to  move  it  in  the  opposite 
direction  about  the  same  axis.  For  let  P  be  any  one  of  such 
a  system  of  pressures,  and  O  a  fixed  axis,  and  OM  perpen- 
dicular to  the  direction  of  P,  then  whatever  may  be  the 
point  of  application  of  P,  the  work  of  that  pressure  is  the 
same  as  though  it  were  applied  at  M.  Suppose  the  whole 
system  to  be  moved  through  an  exceeding  small 
angle  d  about  the  point  O,  and  let  OM  be  repre- 
sented by  Pi  then  will  pQ  represent  the  space 
described  by  the  point  M,  which  will  be  actually 
in  the  direction  of  the  force  P,  therefore  the  work 
of  P=P  .  p  .  6.  Now  let  P,,  P2,  P,,  &c.  represent  those 
pressures  which  act  in  the  direction  of  the  motion,  and  P'x, 
P'2,  &c.  those  which  act  in  the  opposite  direction,  and  let 
Pupvpv  &c.  be  the  perpendiculars  on  the  first,  and  j/l5  p'n 
p '„  &c.  be  the  perpendiculars  on  the  second ;  therefore  by 
the  principle  of  the  equality  of  moments  Pj^  +  P^  +  ^^P* 
•f  &c.  =  P'^  +  P'2y2+  P'8y3  +  &c. ;  therefore  multiplying 
both  sides  by  0,  P  j>^  +  Pj^  +  P3^  =  P',^  +  P'^pV  -f 
P'^V  +  &c. ;  but  Pjj?^,  P',^>'^,  &c.  are  the  works  of  the 
forces  P15  P'15  &c. ;  therefore  the  aggregate  work  of  those 
which  tend  to  move  the  system  in  one  direction  is  equal  to 
the  aggregate  of  those  which  tend  to  move  it  in  the  opposite 
direction. 


64.  THE  ACCUMULATION  OF  WOEK  IN  A  MOVING  BODY. 

In  every  moving  bod^  there  is  accumulated,  by  the  action 
of  the  forces  whence  its  motion  has  resulted,  a  certain 
amount  of  power  which  it  reproduces  upon  any  resistance 
opposed  to  its  motion,  and  which  is  measured  by  the  work 
done  by  it  upon  that  obstacle.  Not  to  multiply  terms,  we 
shall  speak  of  this  accumulated  power  of  working,  thus 
measured  by  the  work  it  is  capable  of  producing,  as  ACCU- 
MULATED WOEK.  It  is  in  this  sense  that  in  a  ball  fired  from 

*  Note  (o)  Ed.  App. 


THE   ACCUMULATION   OF   WORK.  59 

a  cannon  there  is  understood  to  be  accumulated  the  work  it 
reproduces  upon  the  obstacles  which  it  encounters  in  its 
flight ;  that  in  the  water  which  flows  through  the  channel 
of  a  mill  is  accumulated  the  work  which  it  yields  up  to  the 
wheel ;  *  and  that  in  the  carriage  which  is  allowed  rapidly 
to  descend  a  hill  is  accumulated  the  work  which  carries  it  a 
considerable  distance  up  the  next  hill.  It  is  when  the  pres- 
sure under  which  any  work  is  done,  exceeds  the  resistance 
opposed  to  it,  that  the  work  is  thus  accumulated  in  a  moviftg 
body ;  and  it  will  subsequently  be  shown  (Art.  69.)  that  in 
every  case  the  work  accumulated  is  precisely  equal  to  the 
work  done  upon  the  body  beyond  that  necessary  to  over- 
come the  resistances  opposed  to  its  motion,  a  principle 
which  might  almost  indeed  be  assumed  as  in  itself  evident. 

65.  The  amount  of  work  thus  accumulated  in  a  body 
moving  with  a  given  velocity,  is  evidently  the  same,  what- 
ever may  have  been  the  circumstances  under  which  its 
velocity  has  been  acquired.  Whether  the  velocity  of  a  ball 
has  been  communicated  by  projection  from  a  steam  gun,  or 
explosion  from  a  cannon,  or  by  being  allowed  to  fall  freely 
from  a  sufficient  height,  it  matters  not  to  the  result ;  pro- 
vided the  same  velocity  be  communicated  to  it  in  all  three 
cases,  and  it  be  of  the  same  weight,  the  work  accumulated 
in  it,  estimated  by  the  effect  it  is  capable  of  producing,  is 
evidently  the  same. 

In  like  manner,  the  whole  amount  of  work  which  it  is 
capable  of  yielding  to  overcome  any  resistance  is  the  same, 
whatever  may  be  the  nature  of  that  resistance. 

66.    To    ESTIMATE    THE    NUMBER    OF    UNITS    OF    WORK    ACCUMU- 
LATED  IN   A   BODY   MOVING   WITH   A   GIVEN   VELOCITY. 

Let  w  be  the  weight  of  the  body  in  pounds,  and  v  its 
velocity  in  feet. 

Now  suppose  the  body  to  be  projected  with  the  velocity  v 
in  a  direction  opposite  to  gravity,  it  will  ascend  to  the  height 
h  from  which  it  must  have  fallen,  to  acquire  that  same  velo- 
city v  (Art.  47.);  there  must  then  at  the  instant  of  projection 
have  been  accumulated  in  it  an  amount  of  work  sufficient  to 
raise  it  to  this  height  h ;  but  the  number  of  units  of  work 

*  This  remark  applies  more  particularly  to  the  under-shot  wheel,  which  is 
carried  round  by  the  rush  of  the  water. 


60  THE  ACCUMULATION   OF  WORK. 

requisite  to  raise  a  weight  w  to  a  height  A,  is  represented  by 
wh  ;  this  then  is  the  number  of  units  of  work  accumulated 
in  the  body  at  the  instant  of  projection.  But  since  h  is  the 
height  through  which  the  body  must  fall  to  acquire  the  velo- 

city v,  therefore  u2—  %gh  (Art.  47.)  ;  therefore  h=%—  ;  whence 

it  follows  that  if  U  represent  the  number  of  units  of  work 
accumulated, 

nn 


Moreover  it  appears  by  the  last  article  that  this  expression 
represents  the  work  accumulated  in  a  body  weighing  w 
pounds,  and  moving  with  a  velocity  of  v  feet,  whatever  may 
have  been  the  circumstances  under  which  that  velocity  was 
accumulated. 


The  product  —  j'W2  is  called  the  vis  VIVA  of  the  body,  so 
that  the  accumulated  work  is  represented  by  half  the  vis 
viva,  the  quotient  (—  J  is  called  the  MASS  of  the  body.* 

67.  To  estimate  the  work  accumulated  in  a  body,  or  lost  by 
it,  as  it  passes  from  one  velocity  to  another. 

In  a  body  whose  weight  is  w,  and  which  moves  with  a 
velocity  v  there  is  accumulated  a  number  of  units  of  work 

w 
represented  (Art.  66.)  by  the  formula  £—  v\      After  it   has 

passed  from  this  velocity  to  another  V,  there  will  be  accumu- 

w 
lated  in  it  a  number  of  units  of  work,  represented  by  ^  —  V2, 

so  that  if  its  last  velocity  be  greater  than  the  first,  there 
will  have  been  added  to  the  work  accumulated  in  it  a  num- 

ber of  units  represented  by  J—  Y2—  J—  v2;  or  if  the  second 

velocity  be  less  than  the  first,  there  will  have  been  taken 
from  the  work  accumulated  in  it  a  number  of  units  repre- 

sented by  £—  i>2—  J—  V2.      So  that  generally  if  U  represent 

the  work  accumulated  or  lost  by  the  body,  in  passing  from 
the  velocity  v  to  the  velocity  Y,  then 

*  Note  (p),  Ed.  App. 


THE   ACCUMULATION   OF   WOKK.  61 


U=±i-{y-t^  ----  (45), 

where  the  ±  sign  is  to  be  taken  according  as  the  motion  is 
accelerated  or  retarded. 


68.  The  work  accumulated  in  a  body,  whose  motion  is  accele- 
rated through  any  given  space  by  given  forces  is  equal  to 
the  work  which  it  would  he  necessary  to  do  upon  the  body 
to  cause  it  to  move  hack  again  through  the  same  space 
when  acted  upon  by  the  same  forces. 

For  it  is  evident  that  if  with  the  velocity  which  a  body 
has  acquired  through  any  space  AB  by  the 
action  of  any  forces  whose  direction  is  from  A 
towards  B,  it  be  projected  back  again  from  B 
towards  A,  then  as  it  returns  through  each 
successive  small  part  or  element  of  its  path,  it 
will  be  retarded  by  precisely  the  same  forces  as  those  by 
which  it  was  accelerated  when  it  before  passed  through  it  ; 
so  that  it  will,  in  returning  through  each  such  element,  lose 
the  same  portion  of  its  velocity  as  before  it  gained  there  ; 
and  when  at  length  it  has  traversed  the  whole  distance  BA, 
and  reached  the  point  A,  it  will  have  lost  between  B  and  A 
a  velocity,  and  therefore  an  amount  of  work  (Art.  67.), 
precisely  equal  to  that  which  before  it  gained  between  A 
and  B.  Now  the  work  lost  between  B  and  A  is  the  work 
necessary  to  overcome  the  resistances  opposed  to  the  motion 
through  'B  A.  The  work  accumulated  from  A  to  B  is  there- 
fore equal  to  the  work  which  would  be  necessary  to  over- 
come the  resistances  between  B  and  A,  or  which  would  be 
necessary  to  move  the  body  from  a  state  of  rest,  and  with  a 
uniform  motion,  in  opposition  to  these  resistances,  through 
BA.  Let  this  work  be  represented  by  U  ;  also  let  v  be  the 
velocity  with  which  the  body  started  from  A,  and  Y  that 

which  it  has  acquired  at  B.     Then  will  J—  (Y3—  v*)  repre- 
sent the  work  accumulated  between  A  and  B, 


If  the  body,  instead  of  being  accelerated,  had  been 
retarded,  then  the  work  lost  being  that  expended  in  over- 
coming the  retarding  forces,  is  evidently  that  necessary  to 


62  THE   ACCUMULATION    OF   WORK. 

move  the  body  uniformly  in  opposition  to  these  retarding 
forces  through  AB  ;  so  that  if  this  force  be  represented  by 

w 

TJ,  then,  since  -|  —  (V2  —  V2)  is  in  this  case  the  work  lost,  we 

t/ 

shall  have  v*—  Y2=^-.     Therefore,  generally, 

(46), 


where  the  sign  ±  is  to  be  taken  according  as  the  motion  is 
accelerated  or  retarded. 


69.  The  work  accumulated  in  a  body  which  has  moved 
through  any  space  acted  upon  by  any  force,  is  equal  to  the 
excess  of  the  work  which  has  been  done  upon  it  by  those 
forces  which  tend  to  accelerate  its  motion  above  that  which 
has  been  done  upon  it  by  those  which  tend  to  retard  its 
motion. 

For  let  R  be  the  single  force  which  would  at  any  point  P 
(see  last  fig.)  be  necessary  to  move  the  body  back  again 
through  an  exceeding  small  element  of  the  same  path  (the 
other  forces  impressed  upon  it  remaining  as  before) ;  then  it 
follows  by  Art.  54:.  that  the  work  of  R  over  this  element  of 
the  path  is  equal  to  the  excess  of  the  work  over  that 
element  of  the  forces  which  are  impressed  upon  the  body  in 
the  direction  of  its  motion  above  the  work  of  those 
impressed  in  the  opposite  direction.  Now  this  is  true  at 
every  point  of  the  path ;  therefore  the  whole  work  of  the 
force  II  necessary  to  move  the  body  back  again  from  B  to  A 
is  equal  to  the  excess  of  the  work  done  upon  it,  by  the 
impressed  forces  in  the  direction  of  its  motion,  #bove  the 
work  done  upon  it  by  them  in  a  direction  opposed  to  its 
motion ;  whence  also  it  follows,  by  the  last  proposition,  that 
the  accumulated  work  is  equal  to  this  excess.  There- 
fore, &c. 

*70.  If  P  represent  the  force  in  the  direction  of  the 
motion  which  at  a  given  distance  S,  measured  along  the 
path,  acts  to  accelerate  the  motion  of  the  body,  this  force 
being  understood  not  to  be  counteracted  by  any  other,  or  to 
be  the  surplus  force  in  the  direction  of  the  motion  over  and 


THE   ACCUMULATION   OF   WORK.  (>3 

above  any  resistance  opposed  to  it,  then  will  f  PdS  be  the 

o 

work  which  must  be  done  in  an  opposite  direction  to  over- 

s 
come  this  force  through  the  space  S,  or  U—  f 

o 


by  equation  (46),  Y2-V:=±—      -  ......  (47). 


71.  If  the  force  P  tends  at  first  towards  the  direction  in 
which  the  body  moves,  so  as  to  accelerate  the  motion,  and 
if  after  a  certain  space  has  been  described  it  changes  its 
direction  so  as  to  retard  the  motion,  and  U1  represent  the 
value  of  U  in  respect  to  the  former  motion,  and  Yx  the 
velocity  acquired  when  that  motion  has  terminated,  whilst 
U2  is  the  value  of  U  in  respect  to  the  second  or  retarded 
motion,  and  if  v  be  the  initial  and  Y  the  ultimate  or  actual 
velocity,  then 


V-V- 

"    " 


W  ' 

.  ;  .  .  .  .  (48). 


As  U2  increases,  the  actual  velocity  Y  of  the  body  con- 
tinually diminishes  ;  and  when  at  length  U^Uj,  that  is 
when  the  whole  work  done  (above  the  resistances)  in  a 
direction  opposite  to  the  motion,  comes  to  equal  that  done, 
before,  in  the  direction  of  the  motion,  then  Y=v,  or  the 
velocity  of  the  body  returns  again  to  that  which  it  had 
when  the  force  P  begjan  to  act  upon  it.  This  is  that  gene- 
ral case  of  reciprocating  motion  which  is  so  frequently  pre- 
sented in  the  combinations  of  machinery,  and  of  which  the 
crank  motion  is  a  remarkable  example. 


If  the  force  which  accelerates  the  body's  motion  act 
always  towards  the  same  centre  S,  and  SJ  be  taken  equal  to 


64:  THE   ACCUMULATION   OF   WOKK. 

SB,  it  has  been  shown  (Art  55.)  that  the  work 
necessary  to  move  the  body  along  the  curve  from 
B  to  A,  'is  equal  to  that  which  would  be  necessary 
to  move  it  through  the  straight  line  &A.  The 
accumulated  work  is  therefore  equal  to  that  neces- 
sary to  move  the  body  through  the  difference  5A 
of  the  two  distances  SA  and  SB  (Art.  68.).  If  these 
distances  be  represented  by  3^  and  Ra,  and  P 
represent  the  pressure  with  which  the  body's  motion  along 
JA  would  be  resisted  at  any  distance  R  from  the  point  S, 

RI 
then  /  ~PdH  will  represent  this  work.     Moreover  the  work 

R<2 

accumulated  in  the  body  between  A  and  B  is  represented 

by  -J (Y2 — v*),  if  Y  represent  the  velocity  at  B  and  v  that 

at  A, 


73.  The  work  accumulated  in  the  body  while  it  descends 
the  curve  AB,  is  the  same  as  that  which  it  would  acquire  in 
falling  directly  towards  S  through  the  distance  A5,  for  both 
of  these  are  equal  to  the  work  which  w^ould  be  necessary  to 
raise  the  body  from  b  to  A.  Since  then  the  work  accumu- 
lated by  the  body  through  AB  is  equal  to  that  which  it  would 
accumulate  if  it  fell  through  A5,  it  follows  that  velocity 
acquired  by  it  in  falling,  from  rest,  through  AB  is  equal  to 
that  which  it  would  acquire  in  falling  through  AZ>.  For  if 
Y  represent  the  velocity  acquired  in  the  one  case,  and  Yt 
that  in  the  other,  then  the  accumulated  work  in  the  first  case 

W  W 

is  represented  by  $  — Ya,  and  that  in  the  second  case  by  -J- —  Y!  a, 

W  W 

therefore  i  —  Ya  =  J  — Y,8,  therefore  Y= V,. 

From  this  it  follows,  that  if  a  body  descend,  being  pro- 
jected obliquely  into  free  space,  or  sliding  from  rest  upon 
any  curved  surface  or  inclined  plane,  and  be  acted  upon  only 
by  the  force  of  gravity  (that  is,  subject  to  no  friction  or 
resistance  of  the  air  or  other  retarding  cause),  then  the  velo- 


TIIE   ACCUMULATION   OF   WORK.  65 

city  acquired  by  it  in  its  descent  is  precisely  the  same  as 
though  it  had  fallen  vertically  through  the  same  height. 

74.  DEFINITION.  The  ANGULAR  VELOCITY  of  a  body  which 
rotates  about  a  fixed  axis  is  the  arc  which  every  particle  of 
the  body  situated  at  a  distance  unity  from  the  axis  describes 
in  a  second  of  time,  if  the  body  revolves  uniformly  ;  or,  if 
the  body  moves  with  a  variable  motion,  it  is  the  arc  which  it 
would  describe  in  a  second  of  time  if  (from  the  instant  when 
its  angular  velocity  is  measured)  its  revolution  were  to 
become  uniform. 


75.    THE   ACCUMULATION   OF   WORK   IN   A   BODY   WHICH, 
ROTATES   ABOUT   A  FIXED   AXIS. 

Propositions  68  and  69  apply  to  every  case  of  the  motion 
of  a  heavy  body.  In  every  such  case  the  work  accumulated 
or  lost  by  the  action  of  any  moving  force  or  pressure,  whilst 
the  body  passes  from  any  one  position  to  another,  is  equal 
to  the  work  which  must  be  done  in  an  opposite  direction,  to 
cause  it  to  pass  back  from  the  second  position  into  the  first. 
Let  us  suppose  U  to  represent  this  work  in  respect  to  a  body 
of  any  given  dimensions,  which  has  rotated  about  a  fixed 
axis  from  one  given  position  into  another,  by  the  action  of 
given  forces. 

Let  «,  be  taken  to  represent  the  ANGULAR  VELOCITY  of  the 
body  after  it  has  passed  from  one  of  these  positions  into 
another.  Then  since  a  is  the  actual  velocity  of  a  particle  at 
distance  unity  from  the  axis,  therefore  the  velocity  of  a  par- 
ticle at  any  other  distance  pt  from  the  axis  is  ap,.  Let  j* 
represent  the  weight  of  each  unit  of  the  volume  of  the  body, 
and  mt  the  volume  of  any  particle  whose  distance  from  the 
axis  is  pj,  then  will  the  weight  of  that  particle  be  pml ;  also 
its  velocity  has  been  shown  to  be  ap15  therefore  the  amount 
of  work  accumulated  in  that  particle  is  represented  by 

-^,  or  by  ia2 


Similarly  the  different  amounts  of  work  accumulated  in 
the  other  particles  or  elements  of  the  body  whose  distances 
from  the  axis  are  represented  by  p2,  p3,  .  .  .  and  their 

i* 

volumes  by  m2,  ma,  ra<  .  .  .  .,  are  represented  by  ia2-^apaa, 

y 

£a3-  77^p33,  &c. ;  so  that  the  whole  work  accumulated  is  repre- 
y 


66  ANGULAR   VELOCITY. 


U»  Ui  '  ttt 

eented  by  the  sum  ia2-m1p12+ -Ja2-m2p22 +-|a2-m3p8s  +   . 

y  .   y  ff 

,  or  by  £a2-  {m1p12+m2p22+m3p32+ }. 

The  sum  m1  ??  +  m2  p22  +  mg  p32  +  .  .  .  .,  or  2mp2  taken  in 
respect  to  all  the  particles  or  elements  which  compose  the 
body,  is  called  its  MOMENT  OF  INERTIA  in  respect  to  the 
particular  axis  about  which  the  rotation  takes  place.  Let  it 

be  represented  by  I ;   then  will  -Ja2  .  I — J  .  I,  represent  the 

whole  amount  of  work  accumulated  in  the  body  whilst  it  has 
been  made  to  acquire  the  angular  velocity  a  from  rest.  If 
therefore  U  represent  the  work  which  must  be  done  in  an 
opposite  direction  to  cause  the  body  to  pass  back  from  its 
last  position  into  its  first, 


=#  )? 


If  instead  of  the  body's  first  position  being  one  of  rest,  it 
had  in  its  first  position  been  moving  with  an  angular  velocity 
«x  which  had  passed,  in  its  second  position,  into  a  velocity 
a  ;  and  if  U  represent,  as  before,  the  work  which  must  be 
done  in  an  opposite  direction,  to  bring  this  body  back  from 


its  second  into  its  first  position,  then  is  -Ja2  (  -j  I  —  %a*  (  -j  I, 
or  J  (  -  )  (a2  —  axs)  I,  the  work  accumulated  between  the  first 


9 

and  second  positions  ;  therefore 


(«'-<)!=  ±TI, 


where  the  sign  ±  is  to  be  taken  according  as  the  motion  is 
accelerated  or  retarded  between  the  first  and  second  posi- 
tions, since  in  the  one  case  the  angular  velocity  increases 
during  the  motion,  so  that  a2  is  greater  than  a^,  whilst  in  the 
latter  case  it  diminishes,  so  that  a2  is  less  than  a^. 

76.   If  during  one  part  of  the  motion,  the  work  of  the 


ANGULAR   VELOCITY.  67 

impressed  forces  tends  to  accelerate,  and  during  another  to 
retard  it,  and  the  work  in  the  former  case  be  represented  by 
Uj,  and  in  the  latter  by  U9,  then 


From  this  equation  it  follows  that  when  112=11,,  or  when 
the  work  U2  done  by  the  forces  which  tend  to  resist  the 
motion  at  length,  equals  that  done  by  the  forces  which  tend 
to  accelerate  the  motion,  then  a— an  or  the  revolving  body 
then  returns  again  to  the  angular  velocity  from  which  it  set 
out.  "Whilst,  if  UV *  never  becomes  equal  to  U,  in  the  course 
of  a  revolution,  then  the  angular  velocity  a  does  not  return 
to  its  original  value,  but  is  increased  at  each  revolution ; 
and  on  the  other  hand,  if  U2  becomes  at  each  revolution 
greater  than  U^,  then  the  angular  velocity  is  at  each  revolu- 
tion diminished. 

The  greater  the  moment  of  inertia  I  of  the  revolving 
mass,  and  the  greater  the  weight  f*  of  its  unit  of  volume 
(that  is,  the  heavier  the  material  of  which  it  is  formed),  the 
less  is  the  variation  produced  in  the  angular  velocity  a  by 
any  given  variation  of  II  or  U1— U2  at  different  periods  of 
the  same  revolution,  or  from  revolution  to  revolution ;  that 
is,  the  more  steady  is  the  motion  produced  by  any  variable 
action  of  the  impelling  force.  It  is  on  this  principle  that 
the  fly-wheel  is  used  to  equalize  the  motion  of  machinery 
under  a  variable  operation  of  the  moving  power,  or  of  the 
resistance.  It  is  simply  a  contrivance  for  increasing  the 
moment  of  inertia  of  the  revolving  mass,  and  thereby 
giving  steadiness  to  its  revolution,  under  the  operation  of 
variable  impelling  forces,  011  the  principles  stated  above. 
This  great  moment  of  inertia  is  given  to  the  fly-wheel,  by 
collecting  the  greater  part  of  its  material  on  the  rim,  or 
about  the  circumference  of  the  wheel,  so  that  the  distance 
p  of  each  particle  which  composes  it,  from  the  axis  about 
which  it  revolves,  may  be  the  greatest  possible,  and  thus 
the  sum  2mp2,  or  I,  may  be  the  greatest  possible.  ^  At  the 
same  time  the  greatest  value  is  given  to  the  quantity  f*,  by 
constructing  the  wheel  of  the  heaviest  material  applicable 
to  the  purpose. 

What  has  here  been  said  will  best  be  understood  in  its 
application  to  the  CRANK. 


68  ANGULAH   VELOCITY. 

77.  If  we  conceive  a  constant  pressure  Q  to  act  upon  the 
B  arm  CB  of  the  crank 

... "X^^^^P  in  tne  direction  AB  of 

/      VVCi    >^::^3jL  ^6    crank   ro(^?    an^    a 

I    c@r     j  t  '^ ^-^r~  constant   resistance   R 

/y*  to   be   opposed  to  the 

revolution  of  the  axis 

C  always  at  the  same  perpendicular  distance  from  that  axis, 
it  is  evident  that  since  the  perpendicular  distance  at  which 
Q  acts  from  the  axis  is  continually  varying  (being  at  one 
time  nothing,  and  at  another  equal  to  the  whole  length  CB 
of  the  arm  of  the  crank),  the  effective  pressure  upon  the 
arm  CB  must  at  certain  periods  of  each  revolution  exceed  the 
constant  resistance  opposed  to  the  motion  of  that  arm,  and 
at  other  periods  fall  short  of  it ;  so  that  the  resultant  of 
this  pressure  and  this  resistance,  or  the  unbalanced  pressure 
P  upon  the  arm,  must  at  one  period  of  each  revolution  have 
its  direction  in  the  direction  of  the  motion,  and  at  another 
time  opposite  to  it.  Representing  the  work  done  upon  the 
arm  in  the  one  case  by  tF,,  and  in  the  other  by  U2,  it  follows 
that  if  U^TJ,  the  arm  will  return  in  the  course  of  each 
revolution,  from  the  velocity  which  it  had  when  the  work 
Uj  began  to  be  done,  to  that  velocity  again  when  the  work 
U2  is  completed.  If  on  the  contrary  \Jl  exceed  Ua,  then  the 
velocity  will  increase  at  each  revolution  ;  and  if  Uj  be  less 
than  u  2,  it  will  diminish.  It  is  evident  from  equation  (52), 
that  the  greater  the  moment  of  inertia  I  of  the  body  put 
in  motion,  and  the  greater  the  weight  M-  of  its  unit  of 
volume,  the  less  is  the  variation  in  the  value  of  a,  pro'duced 
by  any  given  variation  in  the  value  of  Uj — U2 ;  the  less 
therefore  is  the  variation  in  the  rotation  of  the  arm  of  the 
crank,  and  of  the  machine  to  which  it  gives  motion,  pro- 
duced by  the  varying  action  of  the  forces  impressed  upon  it. 
Now  the  fly-wheel  being  fixed  upon  the  same  axis  with  the 
crank  arm,  and  revolving  with  it,  adds  its  own  moment  of 
inertia  to  that  of  the  rest  of  the  revolving  mass,  thereby 
increasing  greatly  the  value  of  I,  and  therefore,  on  the  prin- 
ciples stated  above,  equalizing  the  motion,  whilst  it  does  not 
otherwise  increase  the  resistance  to  be  overcome,  than  by 
the  friction  of  its  axis,  and  the  resistance  which  the  air 
opposes  to  its  revolution.* 

*  We  shall  hereafter  treat  fully  of  the  crank  and  fly-wheel 


ANGULAR   VELOCITY.  69 

78.  The  rotation  of  a  T)ody  about  a  fixed  axis  when  acted 
upon  ly  no  other  moving  force  than  its  weight. 

Let  U  represent  the  work  necessary  to  raise  it  from  its 
second  position  into  the  first  if  it  be  descending,  or  from  its 
first  into  its  second  position  if  it  be  ascending,  and  let  «x  be 
its  angular  velocity  in  the  first  position,  and  a  in  the  second ; 

•fT>m->    ~V\-tr   Ck/rna-f-i  r\ri    ('\~\\ 


then  by  equation  (51), 


"Now  it  has  been  shown  (Art.  60.),  that  the  work  necessary 
to  raise  the  body  from  its  second  position  into  the  first  if  it 
be  descending,  or  from  its  first  into  its  second  if  it  be 
ascending  (its  weight  being  the  only  force  to  be  overcome), 
is  the  same  as  would  be  necessary  to  raise  its  whole  weight 
collected  in  its  centre  of  gravity  from  the  one  position  into 
the  other  position  of  its  centre  of  gravity.  Let  CA  repre- 
sent the  one,  and  CA1  the  other  position  of 
the  body,  and  G  and  G^  the  two  correspond- 
ing positions  of  the  centre  of  gravity,  then 
will  the  work  necessary  to  raise  the  body 
from  its  position  CA  to  its  position  CA1?  be 
equal  to  that  which  is  necessary  to  raise  its 
whole  weight  "W,  supposed  collected  in  G, 
from  that  point  to  Gj  ;  which  by  Article  56,  is  the  same  as 
that  necessary  to  raise  it  through  the  vertical  height  GM. 

Let  now  CG=CG,=:A,  let  CD  be  a  vertical  line  through 
C,  let  G.CD—  ^  and  GCD=d,  in  the  case  in  which  the 
body  descends,  and  conversely  when  it  ascends;  therefore 
GM=KN"1=CK—  CN^h  cos.  4—  h  cos.  ^  when  the  body 
descends,  or  =h  cos.  0X  —  h  cos.  6  when  it  ascends  from  the 
position  AC  to  AC1?  since  in  this  last  case  GCD=^  and 
Q1GD=6.  Therefore  GM=±A  (cos.  A—  cos.  6\  the  sign  ± 
being  taken  according  as  the  body  ascends  or  descends. 
U=W  .  GM=±WA  cos.  4  —  cos. 


.'.  by  equation  (51)  a*=ai,*  +  {  —  3—\  (cos.  6  —  cos.  ^). 
If  M  represent  the  volume  of  the  revolving  body  Mf*=  W, 
(cos.  d-cos.  ^)  .....  (53). 


When  the  body  has  descended  into  the  vertical  position, 


TO  MOMENT   OF   INEKTIA. 

0=0,  so  that  (cos.  6  —  cos.  ^)=1  —  cos.  ^=2  sin.2-^.     When 
it  has  ascended  into  that  position  d=tf,  so  that  (cos.  6  —  cos. 
*,)=—  (1  +  cos.  ^)=—  2  cos.2^. 
In  the  first  case,  therefore, 


In  the  second  case, 

e.'tt  .....  (55). 


When  the  body  has  descended  or  ascended  into  the  hori- 

if 
zontal  position  d=~,  so  that  (cos-  ^  —  cos-  ^)—  —  cos-  ^i-    But 

it  is  to  be  observed,  that  if  the  body  have  descended  into 

the  horizontal  position,  0,  must  have  been  greater  than  ^, 

2 

and  therefore  cos.  6l  must  be  negative  and  equal  to  —  cos. 
BCG,  ;  so  that  if  we  suppose  6t  to  be  measured  from  CB  or 
CD,  according  as  the  body  descends  or  ascends,  then  (cos. 
6  —  cos.  fl,)=±cos.  d1?  and  we  have  for  this  case  of  descent 
or  ascent  to  a  horizontal  position 


(56.) 


If  the  body  descend  from  a  state  of  rest,  al=0. 
.-.  by  equation  (53)  <*?=-$—  (cos.  d—  cos.  d,)  .  .  .  (57). 

Thus  the  angular  velocity  acquired  from  rest  is  less  as  the 
moment  of  inertia  I  is  greater  as  compared  with  the  volume 
M,  or  as  the  mass  of  the  body  is  collected  farther  from  its 
axis. 


THE  MOMENT  OF  INERTIA. 

79.  Having  given  the  moment  of  inertia  of  a  ~body,  or  system 
of  bodies,  about  an  axis  passing  through  its  centre  of 
gravity,  to  find  its  moment  of  inertia  about  an  axis,  par- 
allel to  the  first,  passing  through  any  other  point  in  the 
body  or  system. 

Let  m,  be  any  element  of  the  body  or  system,  7 


MOMENT    OF    INERTIA.  71 

plane  perpendicular  to  the  axis,  about 
which  the  moments  are  to  be  measured,  A 
\  the  point  where  this  plane  is  intersected 
by  that  axis,  and  G  the  point  where  it  is 
intersected  by  the  parallel  axis  passing 
through  the  centre  of  gravity  of  the  body.  Join  AG, 
A.m^  Gra,,  and  draw  m1M1  perpendicular  to  AG.  Let 

JSTow  (Euclid,  2—12.),  Am'  =  AG^+G^'+aAG  .  GM^, 

if  therefore  the  volume  of  the  element  be  represented  by 
ma  and  both  sides  of  the  above  equation  be  multiplied  by  it, 
pj'mj = ffm^ + r12m1  -f  2  Aa^m^ 

And  if  m2,  m^  m4,  &c.  represent  the  volumes  of  any  other 
elements,  and  ps,  r2,  a?2 ;  Pa,  r3,  a?3,  &c.  be  similarly  taken  in 
respect  to  those  elements,  then, 


Adding  these  equations  we  have,  p19m1  +  p22m2  +  p*ma  + 
,+  m3  + 


or 

~Now  2xm  is  the  sum  of  the  moments  of  all  the  elements 
of  the  body  about  a  plane  perpendicular  to  AG,  and  passing 
through  the  centre  of  gravity  G  of  the  body.  Therefore 
(Art.  IT.) 


Also  2pV&  is  the  moment  of  inertia  of  the  body  about  the 
given  axis  passing  through  A,  and  2r2m  is  the  moment  of 
inertia  about  an  axis  parallel  to  this,  passing  through  the 
centre  of  gravity  of  the  body.  Let  the  former  moment  be 
represented  by  X  ;  and  the  latter  by  I  ;  and  let  the  volume 
of  the  body  ^m  be  represented  by  M, 

.-.  ];=AfM+I  .....  (58). 

From  which  relation  the  moment  of  inertia  (I,)  about  any 
axis  may  be  found,  that  (I)  about  an  axis  parallel  to  it,  and 
passing  through  the  centre  of  gravity  of  the  body  being 
known. 

80.  THE  RADIUS  OF  GYRATION.  If  we  suppose  ^  to  be  the 
distance  from  the  axis  passing  through  A,  at  which  distance, 


72  MOMENT   OF   INERTIA. 

if  the  whole  mass  of  the  body  were  collected,  the  moment  of 
inertia  would  remain  the  same,  so  that  &t*M=I,,  then  \  is 
called  the  RADIUS  OF  GYRATION,  in  respect  to  that  axis. 

If  k  be  the  radius  of  gyration,  similarly  taken  in  respect 
to  the  axis  passing  through  G1?  so  that  &aM=I,  then,  substi 
tuting  in  the  preceding  equation,  and  dividing  by  M, 


The  following  are  examples  of  the  determination  of  the 
moments  of  inertia  of  bodies  of  some  of  the  more  common 
geometrical  forms,  about  the  axes  passing  through  their  cen- 
tres of  gravity :  they  may  thence  be  found  about  any  other 
axes  parallel  to  these,  by  equation  (58). 


1.  The  moment  of  inertia  of  a  thin  uniform  rod  about  an 
axis  perpendicular  to  its  length  and  passing  through  its 
middle  point. 

Let  m  represent  an  element  of  the  rod  contained  between 
two  plane  sections  perpendicular  to  its 
faces,  the  area  of  each  of  which  is  «,  and 
whose  distance  from  one  another  is  Ap, 
I  and  let  K  and  Ap  be  so  small  -that  every 

point  in  this  element  may  be  considered  to  be  at  the  same 
distance  p  from  the  axis  A,  about  which  the  rod  revolves. 
Then  is  the  volume  of  the  element  represented  by  «Ap?  and 
its  moment  of  inertia  about  A  by  «p2Ap.  So  that  the  whole 
moment  of  inertia  I  of  the  bar  is  represented  by  2/cpaAp,  or, 
since  K  is  the  same  throughout  (the  bar  being  uniform),  by 
«;2paAp;  or  since  Ap  is  infinitely  small,  it  is  represented  by 


the  definite  integral  KJ  p2<#p,  where  I  is  the  whole  length 


—  \i 
of  the  bar, 


or  I=TV^3  .....  (60). 


*82.  The  moment  of  inertia  of  a  thin  rectangular  lamina 
about  an  axis,  passing  through  its  centre  of  gravity r,  and 
parallel  to  one  of  its  sides. 

It  is  evident  that  such  a  lamina  may  be  conceived  to  be 


MOMENT    OF   INERTIA. 


made  up  of  an  infinite  number  of  slendei 
rectangular  rods  of  equal  length,  each  of 
which  will  be  bisected  by  the  axis  AB, 
and  that  the  moment  of  inertia  of  the 
whole  lamina  is  equal  to  the  sum  of  the 
moments  of  inertia  of  these  rods.  Now  if  K  be  the  section 
of  any  rod,  and  I  the  length  of  the  lamina,  then  the  moment 
of  inertia  of  that  rod  is,  by  the  last  proposition,  represented 
by  TV/^3 ;  so  that  if  the  section  of  each  rod  be  the  same,  and 
they  be  n  in  number,  then  the  whole  moment  of  inertia  of 
the  lamina  is  -j?nicF.  Now  UK,  is  the  area  of  the  transverse 
section  of  the  lamina,  which  may  be  represented  by  K,  so 
that  the  moment  of  inertia  of  the  lamina  about  the  axis  AB 
is  represented  by  the  formula 

(61). 


*83.  The  moment  of  inertia  of  a  rectangular  parallelopipe- 
don  about  an  axis,  passing  through  its  centre  of  gravity, 
and  parallel  to  either  of  its  edges. 

Let  CD  be  a  rectangular  parallelopipedon,  and  AB  an 
axis  passing  through  its  centre  of  gravity  and 
parallel  to  either  of  its  edges ;  also  let  ab  be 
an  axis  parallel  to  the  first,  passing  through 
the  centre  of  gravity  of  a  lamina  contained 
by  planes  parallel  to  either  of  the  faces  of  the 
parallelepiped.  Let  a,  5,  c,  represent  the 
three  edges  ED,  EF,  EG,  of  the  parallele- 
piped, then  will  the  moment  of  inertia  of  the  lamina  about 
the  axis  ab  be  represented  by  TyK&8,  where  K  is  the  trans- 
verse section  of  the  lamina  (equation  61).  Now  let  the 
perpendicular  distance  between  the  two  axes  AB  and  ab  be 
represented  by  x.  Then  (by  equation  58)  the  moment  of 
inertia  of  the  lamina  about  the  axis  AB  is  represented  by 
the  formula  o^M+y^KJ3,  where  M  represents  the  volume  of 
the  lamina.  Let  the  thickness  of  the  lamina  be  represented 
by  AX  ;  .*.  M  —  ab^x,  K  —  a&x ;  .*.  m*  ina  of  lama  =  dbx*&x  + 
•jijfl&'Aaj ;  .-.  whole  m*  ina  of  parallelepiped  =  ab'Zx'Ax  + 
jJyd^'SAoj ;  or  taking  &x  infinitely  small,  and  representing  the 
moment  of  inertia  of  the  parallelepiped  by  I. 


/»  +  j-c  /*  +|o 

! — abj  x*dx + j-sCtb'J  dx ; 

~\c  — |o 


MOMENT   OF   INEKTIA. 

or  1= 


(62). 


moment  of  inertia  of  an  upright  triangular  2^1 
about  a  vertical  axis  passing  through  its  centre  of  gravity. 

Let  AB  be  a  vertical  axis  passing  through  the  centre  of 
gravity  of  a  prism,  whose  horizontal  section  is 
an  isosceles  triangle  having  the  equal  sides  ED 
and  EF. 

Let  two  planes  be  drawn  parallel  to  the  face 
DF  of  the  prism,  and  containing  between  them 
a  thin  lamina  pq  of  its  volume.  Let  Cm,  the 
perpendicular  distance  of  an  axis  passing  through 
the  centre  of  gravity  of  this  lamina  from  the 
axis  AB,  be  represented  by  a?  ;  also  let  A#  represent  the 
thickness  of  the  lamina. 

Let  DF=  #,  DG  =  5,  and  let  the  perpendicular  from  the 
vertex  E  to  the  base  DF  of  the  triangle  DEF  be  represented 


:.pq  =  -  (fc—  a?)  ;   also  transverse  section  K  of  lamina  = 
c 

.'.volume  M  of  lamina  =  -  (ftf—  oc)&oc.     Therefore  by  equa- 

c 

tions  (58)  and  (61), 

m*  ina  of  lama  about  AB=^(fc—  x)x'Ax+7^3(%c-x)*Ax-, 

c  c 

:.  m*  ina  of  prism  about 
ob 


C         -*c 

Performing  the  integrations  here  indicated,  and  represent- 
ing the  inertia  of  the  prism  about  AB  by  I,  we  have 

......  (63). 


MOMENT   OF   LNEETIA. 


*85.  The  moment  of  inertia  of  a  solid  cylinder  about  its 
axis  of  symmetry. 

Let  AB  be  the  axis  of  such  a  cylinder,  whose  radius  AC 
is  represented  by  #,  and  its  height  by  b.  Con- 
ceive the  cylinder  to  be  made  up  of  cylindrical 
rings  having  the  same  axis  ;  let  AP—  p  be  the 
internal  radius  of  one  of  these,  and  let  its  thick- 
ness PQ  be  represented  by  Ap,  so  that p+ Apis 
D  the  exteral  radius  AQ  of  the  ring.  Then  will 
the  volume  of  the  ring  be  represented  by 
tf£(p-|-Ap)2 — <rrZ>p2,  or  by  tf#[2pAp-|-(Ap)2]  ;  or  if  Ap 
be  taken  exceedingly  small,,  so  that  (Ap)2  may  vanish  as  com- 
pared with  2pAp,  then  is  the  volume  of  the  ring  represented 
by  2<n$pAp. 

Now  this  being  the  case,  the  ring  may  be  considered  as  an 
element  AM  of  the  volume  of  the  solid,  every  part  of  which 
element  is  at  the  same  distance  p  from  the  axis  AB,  so  that 
the  whole  moment  of  inertia  2p2AM  of  the  cylinder  = 


*86.  The  moment  of  inertia  of  a  hollow  cylinder  about 
axis  of  symmetry. 

be  the  external  radius  AC,  and  #2  the  internal 
radius  AP,  and  b  the  height  of  the  cylinder ; 
then  by  the  last  proposition  the  moment  of  in- 
ertia of  the  cylinder  CD,  if  it  were  solid,  would 
be  ^~ba* ;  also  the  moment  of  inertia  of  the 
cylinder  PK,  which  is  taken  from  this  solid  to 
form  the  hollow  cylinder,  would  be  %*la*.  Now 
let  I  represent  the  moment  of  inertia  of  the  hol- 
low cylinder  CP,  therefore 


Let  the  thickness  a,  —  a,  of  the  hollow  cylinder  be  repre- 


sented by  c,  and  its  mean  radius 


therefore 


76  MOMENT   OF   INERTIA.       , 

Substituting  these  values  in  the  preceding  equation,  we  ob  • 
tain 

(65). 


\1.  The  moment  of  inertia  of  a  cylinder  about  an  axis 
passing  through  ^ts  centre  of  gravity,  and  perpendicular 
to  its  axis  of  symmetry. 

Let  AB  be  such  an  axis,  and  let  PQ  represent  a  lamina 
contained  between  planes  perpendicular  to 
this  axis,  and  exceedingly  near  to  each  other. 
Let  CD,  the  axis  of  the  cylinder,  be  repre- 
,  sented  by  5,  its  radius  by  tz,  and  let  CM=a?. 
Take  &x  to  represent  the  thickness  of  the 
lamina,  and  let  MP=y.  Now  this  lamina 
may  be  considered  a  rectangular  parallelo 
piped  traversed  through  its  centre  of  gravity  by  the  axis  AB ; 
therefore  by  equation  (62)  its  moment  of  inertia  about  that  axis 
is  represented  by  i-VC^O^O^)  ft*  +  (^2/)2 }  =y&  \b*y  +  ty*\  &x. 
Now  the  whole  moment  of  inertia  I  of  the  cylinder  about 
AB  is  evidently  equal  to  the  sum  of  the  moments  of  inertia 
of  all  such  laminae  ; 

.'.1= 

Also,  since  x  and  y  are  the  co-ordinates  of  a  point  in  a 
circle  from  its  centre,  therefore  y—  ($2— a?2)*.  Substituting 
this  value  of  y,  and  integrating  according  to  the  well  known 
rules  of  the  integral  calculus,*  we  have 


*88.  The  moment  of  inertia  of  a  cone  about  its  axis  of 
symmetry. 

The  cone  may  be  supposed  to  be  made  up  of  laminae,  such 
as  PQ,  contained  by  planes  perpendicular  to 
the  axis  of  symmetry  AB,  and  each  having  its 
centre  of  gravity  in  that  axis.  Let  BP—  #?,  and 
let  Aa?  represent  the  thickness  of  the  lamina, 
and  y  its  radius  PR.  Then,  since  it  may  be 
considered  a  cylinder  of  very  email  height,  its 
moment  of  inertia  about  AB  (equation  64)  is 
represented  by  fyty^x.  Now  the  moment  of 


Church's  Diff.  and  Intcg.  Calculus,  Arts.  148,  149. 


MOMENT   OF   INEKTIA.  77 

inertia  I  of  the  whole  cone  is  equal  to  the  sum  of  the  mo- 
ments of  all  such  elements, 


Let  the  radius  of  the  base  of  the  cone  be  represented  by 

and  its  height  by  5  ;  therefore- =-,  therefore  Aa?=  -Ay  ; 

y    a  a 


(6T). 


89.  The  moment  of  inertia  of  a  sphere  about  one  of  its 
diameters. 

Let  C  be  the  centre  of  the  sphere  and  AB  the  diameter 
about  which  its  moment  is  to  be  determined. 

eLet  PQ  be  any  lamina  contained  by  planes 
perpendicular  to  AB  ;  let  CM=#,  and  let  &x 
represent  the  thickness  of  the  lamina,  and  y  its 
radius  ;  also  let  CA=a  ;  then  since  this  lamina, 
being  exceedingly  thin,  may  be  considered  a 
cylinder,  its  moment  of  inertia  about  the  axis  AB  is  (equa- 
tion 64)  %*y*&x ;  and  the  moment  of  inertia  I  of  the  whole 
sphere  is  the  sum  of  the  moments  of  all  such  laminae, 


Now  by  the  equation  to  the  circle  y*=a*—  a?2,  therefore 
2/*=&4—  2«V+#4.  If  this  value  be  substituted  for  y4,  and 
the  integration  be  completed  according  to  the  common 
methods,  we  shall  obtain  the  equation, 


(68). 


90.  The  moment  of  inertia  of  a  cone  about  an  axis  gassing 
through  its  centre  of  gravity  and  perpendicular  to  its  axis 
of  symmetry. 

Let  CD  be  an  axis  passing  through  the  centre  of  gravity 


78  MOMENT    OF   INERTIA. 

G  of  the  cone,  and  perpendicular  to  its  axis  of 
symmetry,  and  let  GP  the  distance  of  the  lamina 
from  G,  measured  along  the  axis,  be  represented 
by  x  ;  also  let  the  thickness  of  the  lamina  be  re- 
presented by  &x.  Now  this  lamina  may  be  con- 
sidered a  cylinder  of  exceedingly  small  thick- 
ness. If  its  radius  be  represented  by  y,  its  mo- 
ment of  inertia  about  an  axis  parallel  to  CD  passing  through 
its  centre,  is  therefore  (equation  66)  represented  by 
J^y2  jyQ+-J(Aa?)2}Aa??  or  if  ACC  be  assumed  exceedingly  email, 
it  is  represented  by  fay*&x.  Now  this  being  the  moment  of 
the  lamina  about  an  axis  parallel  to  CD,  passing  through  its 
centre  of  gravity,  and  the  distance  of  this  axis  from  CD  be- 
ing a?,  and  also  the  volume  of  the  lamina  being  flry'Aaj,  it  fol- 
lows (equation  58),  that  the  moment  of  the  lamina  about  CD 
is  represented  by  tfyVAoj+J^Aa?—  *  jyV-f  Jy4J  A#. 

Now  the  moment  I  of  the  whole  cone  about  CD  equals 
the  sum  of  the  moments  of  all  such  elements, 


Now  if  a  be  the  radius  of  the  base  of  the  cone  and  ~b  its 
height,  then  since  BG=f  5, 

j—  x    1}  T> 


91.  The  moment  of  inertia  of  a  segment  of  a  sphere  about 
a  diameter  parallel  to  the  plane  of  section. 

Let  ADBE  represent  any  such  portion  of  a  sphere,  and 

T>  AB  a  diameter  parallel  to  the  plane  of  section. 

/^JlX     Let  CD=tf,  CE=5,  and  let  PQ  be  any  lamina 

p— ^:::::::--'-^    contained  by  planes  parallel  to  the  plane  of 

\ Z'-S- •-"""/*"  section :  let  the  distance  of  the  lamina  from 

C=#,  and  let  its  thickness  be  &x  and  its  radius 

,     Then  considering  it  a  cylinder  of  exceeding  small  thick- 


MOVING   FORCES.  79 

ness,  its  moment  of  inertia  about  an  axis  passing  through  its 
centre  of  gravity  and  parallel  to  AB,  is  represented  (equa- 
tion 66)  by  i^y2fy2  +  i(Aa?)2!  AOJ,  or  (neglecting  powers  of  A# 
above  the  first  by  %*y*&x.  Hence,  therefore,  the  moment  of 
this  lamina  about  the  axis  AB  is  represented  (equation 
58)  by  7r?/2(A2>),'£2  _|_  •J-7ry4A#)  or  by  <7r'ji?/V24-J?/4J  A#?;  now  the 
whole  moment  I  of  inertia  of  ADBE  about  AB  is  evidently 
equal  to  the  sum  of  the  moments  of  all  such  laminae, 


.;    =«     y 

-b 

*=a?—x\  therefore  ^V-f  Jy4=Jj2&V— 3^4  +  ^4j. 
Substituting  this  value  in  the  integral  and  integrating,  we 
have 

-955J* (TO) 


THE  ACCELERATION   OF   MOTION  BY  GIYEN 
MOVING  FOECES. 

92.  IF  the  forces  applied  to  a  moving  body  in  the  direc- 
tion of  its  motion  exceed  those  applied  to  it  in  the  opposite 
direction  (both  sets  of  forces  being  resolved  in  the  direction 
of  a  tangent  to  its  path),  the  motion  of  the  body  will  be  ac- 
celerated ;  if  they  fall  short  of  those  applied  in  the  opposite 
direction,  the  motion  will  be  retarded.  In  either  case  the 
excess  of  the  one  set  'of  forces  above  the  other  is  called  the 
MOVING  FORCE  upon  the  body  :  it  is  measured  by  that  single 
pressure  which  being  applied  to  the  body  in  a  direction  op- 
posite to  the  greater  force,  would  just  balance  it ;  or  which, 
had  it  been  applied  to  the  body  (together  with  the  other 
forces  impressed  upon  it)  when  in  a  state  of  rest,  would  have 
maintained  it  in  that  state  ;  and  which,  therefore,  if  applied 
when  its  motion  had  commenced,  would  have  caused  it  to 
pass  from  a  state  of  variable  to  one  of  uniform  motion.  Thus 
the  moving  force  upon  a  body  which  descends  freely  by  gra- 
vity, is  measured  by  its  weight,  that  is,  by  the  single  force 
which,  being  applied  to  the  body  before  its  motion  had  com- 
menced in  a  direction  opposite  to  gravity,  would  just  have 
supported  it,  and  which  being  applied  to  it  at  any  instant  of 

*  Note  (q)  Ed.  App. 


80  RELATIONS   OF 

its  descent,  would  have  caused  its  motion  at  that  instant  to 
pass  from  a  state  of  variable  to  a  state  of  uniform  motion. 
If  the  resistance  of  the  air  upon  its  descent  be  taken  into 
account,  then  the  moving  force  upon  the  body  at  any  instant 
is  measured  by  that  single  pressure  which,  being  applied  up- 
wards, would,  together  with  the  resistance  of  the  air  at  that 
instant,  just  balance  the  weight  of  the  body. 

A  moving  force  being  thus  understood  to  be  measured  by 
&  pressure*  being  in  fact  the  unbalanced  pressure  upon  the 
moving  body,  the  following  relations  between  the  amount  of 
a  moving  force  thus  measured,  and  the  degree  of  acceleration 
produced  by  it  will  become  intelligible.  These  are  laws  of 
motion  which  have  become  known  by  experiment  upon  the 
motions  of  the  bodies  immediately  around  us,  and  by  obser- 
vation upon  those  of  the  planets. 

93.  "When  the  moving  force  upon  a  body  remains  con- 
stantly the  same  in  amount  (as  measured  by  the  equivalent 
pressure)  throughout  the  motion,  or  is  a  uniform  moving 
force,  it  communicates  to  it  equal  additions  of  velocity  in 
equal  successive  intervals  of  time.     Thus  the  moving  force 
upon  a  body  descending  freely  by  gravity  (measured  by  its 
weight)  being  constantly  the  same  in  amount  throughout  its 
descent  (the  resistance  of  the  air  being  neglected),  the  body 
receives  from  it  equal  additions  of  velocity  in  equal  succes- 
sive intervals  of  time,  viz.  32£  feet  in  each  successive  second 
of  time  (Art.  44.). 

94.  The  increments  of  velocity  communicated  to  equal 
todies  by  unequal  moving  forces  (supposed  uniform  as  above) 
are  to  one  another  as  the  amounts  of  those  moving  forces 
(measured  by  their  equivalent  pressures). 

Thus  let  P  and  P1  be  any  two  unequal  moving  forces  upon 
two  equal  bodies,  and  let  them  act  in  the  directions  in  which 
the  bodies  respectively  move ;  let  them  be  the  only  forces 
tending  to  communicate  motion  to  those  bodies,  and  remain 
constantly  the  same  in  amount  throughout  the  motion.  Also 
let  f  and  ft  represent  the  additional  velocities  which  these 
two  forces  respectively  communicate  to  those  two  equal 
bodies  in  each  successive  second  of  time  ;  then  it  is  a  law  of 
the  motion  of  bodies,  determined  by  observation  and  experi- 
ment, that  P  :  P!  ::/:/;. 

*  Pressure  and  moving  force  are  indeed  but  different  modes  of  the  operation 
of  the  same  principle  of  force. 


PRESSURE   AND   MOTION.  81 

if  one  of  the  moving  forces,  as  for  instance  Pn  be  the 
weight  "W  of  the  body  moved,  then  the  value  /j  of  the 
increment  of  velocity  per  second  corresponding  to  that 
moving  force  is  32 l  (Art.  44.)  represented  by  <?, 


=/ (71). 


95.  If  the  amount  or  magnitude  of  the  moving,  force  does 
not  remain  the  same  throughout  the  motion,  or  if  it  be  at 
variable  moving  force,  then  the  increments,  of  velocity  com- 
municated by  it  in  equal  successive  interval^  of  time  are  not 
equal;  they  increase  continually  if  the  moving  force- 
increases,  and  they  diminish  if  it  diminishes.. 

If  two  unequal  moving  forces,  one  or  both  of  them,,  thus 
variable  in  magnitude,  become  the  moving-  forces-  of'  two 
equal  bodies,  the  additional  velocities  which  they  would 
communicate  in  the  same  interval  of  'time  to  those  bodies, 
if  at  any  period  of  the  motion  from  variable  they  become 
uniform,  are  to  one  another  (Art.  94.)  as  the  respective 
moving  forces  at  that  period  of  the  motion. 

Thus  let  f  and  ft  represent  the  -additional  velocities  which 
would  thus  be  communicated  to  two  equal  bodies  in  one 
second  of  time,  if  at  any  instant  the  pressures  P  and  Px, 
which  are  at  that  instant  the  moving  forces  of  those  bodies, 
were  from  variable  to  become  constant  pressures,  then 
(Art.  94.), 


This  being  true  of  any  two  moving  forces,  is  evidently  true, 
if  one  of  them  become  a  constant  force.  Let  P,  represent 
the  weight  W  of  the  body,  then  will  fv  be  represented! 


Let  the  moving  force  P  be  supposed  to  remain  constant 
during  a  number  of  seconds  or  parts  of  a  second,  repre- 
sented by  A£,  and  let  AY  be  the  increment  of  velocity  in 
the  time  &t  on  this  supposition.  Now  /  represents  the 
increment  of  velocity  in  each  second,  and  AY  the  increment 
of  velocity  in  A£  seconds  :  moreover  the  force  P  is  supposed 
constant  (Tiring  A£,  so  that  the  motion  is  uniformly  accele- 
rated during  that  time  (Art.  44.). 

6 


82  RELATIONS    OF 


"Now  this  is  true  (if  the  supposition,  that  P  remains  constant 
during  the  time  A£?  on  which  it  is  founded,  be  true),  how- 
ever small  the  time  &t  may  be.  But  if  this  time  be 
infinitely  small,  the  supposition  on  which  it  is  founded  is  in 
all  cases  true,  for  P  may  in  all  cases  be  considered  to  remain 
the  same  during  an  infinitely  small  period  of  time,  although 
it  does  not  remain  the  same  during  any  time  which  is  not 

AY    dV 

infinitely  small.      Now  when  &t  is  infinitely  small—  -=—,-  ; 

:  A£         dt 

generally  therefore  f=  -^  . 

If  Y  increase  as  the  time  t  increases,  or  if  the  motion  be 

dV 

accelerated,  then  -^-  is  necessarily  a  positive  quantity.     If, 

on  the  contrary,  Y  diminishes  as  the  time  increases,  then 

dV 

-57  is  negative  ;  so  that,  generally, 


the  sign  ±  being  taken  according  as  the  motion  is  accele- 
rated or  retarded.  Substituting  mis  value  of  f  in  the  last 
proportion  we  have  in  the  case,  in  which  P  represents  a 
variable  pressure, 


The  principles  stated  above  constitute  the  fundamental  rela- 
tions of  pressure  and  motion. 


96.  The  velocity  Y  at  any  instant  of  a  body  moving  with 
a  variable  motion,  being  the  space  which  it  would  describe 
in  a  second  of  time,  if  at  that  instant  its  motion  were  to 
become  uniform,  it  follows,  that  if  we  represent  by  &t  any 
number  of  seconds  or  parts  of  a  second,  beginning  from  that 
instant,  and  by  AS,  the  space  which  the  body  would  describe 

*  Note  (r)  Ed.  App. 


PRESSURE   AND   MOTION. 


in  the  time  A£,  if  its  motion  continued  uniform  from  the  com- 
mencement of  that  time,  then, 


this  is  true  if  the  motion  remain  uniform  during  the 
time  A£,  however  small  that  time  may  be,  and  therefore  if  it 
be  infinitely  small.  But  if  the  time  &t  be  infinitely  small, 
the  motion  does  remain  uniform  during  that  time,  however 
variable  may  be  the  moving  force  ;  also  when  &t  is  infi- 

nitely small,  --3  =  -JTT.     Therefore,  generally, 


The  equations  (73)  and  (74)  are  the  fundamental  equations 
of  dynamics  :  they  involve  those  dynamical  results  which 
have  been  discussed  on  other  principles  in  the  preceding 
parts  of  this  work.* 


THE  DESCENT  OF  A  BODY  UPON  A  CURVE. 

*97.  If  the  moving  force  P  upon  a  ~body  varies  (Rreetly  as  its 
distance  at  any  tome  from  a  given  point  towards  which  it 
falls,  then  the  whole  time  of  the  body's  falling  to  that 
point  will  be  the  same,  whatever  may  be  the  distance  from 
which  it  falls. 

Let  A  be  the  point  from  which  the  body  falls,  and  B  a 
point  towards  which  it  falls  along  the  path 
APB,  which  may  be  either  curved  or  straight ; 
also  let  the  body  be  acted  upon  at  each 
point  P  of  its  path,  by  a  force  in  the  direc- 
tion of  its  path  at  that  point  which  varies  as 

*  Thus  if  the  latter  equation  be  inverted,  and  multiplied  by  the  former,  we 
obtain  the  equation 


m  »t  ya va  = 

which  is  identical  with  equation  (47). 


84  RELATIONS    OF 

its  distance  BP,  measured  along  the  path  from  B  ;  the  time 
of  falling  to  B  will  be  the  same,  whatever  may  be  the  dis- 
tance of  the  point  A  from  which  the  body  falls. 

For  let  BP—  S,  and  let  the  force  impelling  the  body 
towards  B  be  represented  by  dS,  where  c  is  a  constant  quan- 
tity ;  suppose  the  body,  instead  of  falling  from  A  towards 
B,  to  be  projected  with  any  velocity  from  B  towards  A,  and 
let  v  be  the  velocity  acquired  at  ?,  and  Y  that  at  A,  and 
let  BA=S15  then  by  equation  (47), 


Suppose  now  the  velocity  of  projection  from  B  to  have 
been  such  as  would  only  just  carry  the  body  to  A,  so  that 
V=0, 


Now  by  equation  (74), 
dtl 


and  if  £T  represent  the  whole  time  in  seconds  occupied  in 
the  ascent  of  the  body  from  B  to  A, 


2' 

It  is  clear  that  the  time  required  for  the  body's  descent 
from  A  to  B  is  equal  to  that  necessary  for  the  ascent  from 
B  to  A,  so  that  the  whole  time  required  to  complete  the 
ascent  and  descent  is  equal  to  T,  and  is  represented  by  the 
formula 


(T6). 


PKESSUKE   AND   MOTION.  85 

Now  this  expression  does  not  contain  S15  i.  e.  the  distance 
from  which  the  body  falls  to  B  ;  the  time  T  is  the  same 
therefore,  whatever  that  distance  may  be. 

THE  SIMPLE  PENDULUM. 

98.  If  a  heavy  particle  P  ~be  imagined  to  ~be  suspended  from  a 
point  C  ~by  a  thread  without  weight,  and  allowed  to  oscillate 
freely,  but  so  as  to  deviate  out  little  on  either  side  of  the 
vertical,  then  will  its  oscillations,  so  long  as  they  are  thus 
small,  be  performed  in  the  same  time  whatever  their  ampli- 
tudes may  be. 

For  let  the  inclination  PCB  of  CP  to  the  vertical  be  repre- 
sented by  6,  and  let  the  weight  w  of  the  particle 
P,  which  acts  in  the  direction  of  the  vertical  VP, 
be  resolved  into  two  others,  one  of  which  is  in  the 
direction  CP,  and  the  other  perpendicular  to  that 
direction  :  the  former  will  be  wholly  counteracted 
by  the  tension  of  the  thread  CP,  and  the  latter  will 
Jbe  represented  by  w  sin.  VPC=w  sin.  &  ;  and,  act- 
ing in  the  direction  in  which  the  particle  P  moves,  this  will 
be  the  whole  impressed  moving  force  upon  it  (Art.  92.)  Now 
so  long  as  the  arc  $  is  small,  this  arc  does  not  differ  sensibly 
from  its  sine,  so  that  for  small  oscillations  the  impressed  mov- 

ing force  upon  P  is  represented  by  wb,  or  by—  ^--,  or  by  —  , 

I  L 

if  I  represent  the  length  CP  of  the  suspending  thread,  and  S 
the  length  of  the  arc  BP.  Now  in  this  expression  w  and  I 
are  constant  throughout  the  oscillation,  the  moving  force  va- 
ries therefore  as  S.  Hence  by  the  last  proposition,  the  small 
oscillations  on  either  side  of  CB  are  isochronous,  since  so  long 
as  they  are  thus  small,  the  impressed  moving  force  in  the 
direction  of  the  motion  varies  as  the  length  of  the  path  BP 
from  the  lowest  point  B.  Since  in  the  last  proposition  the 
moving  force  was  assumed  equal  to  cS,  and  that  here  it  is 

represented  by  yS,  therefore  in  this  case  <?=y.  Substitut- 
ing this  value  in  equation  (76), 


A  single  particle  thus   suspended  by  a  thread   without 


86  THE   PARALLELOGRAM   OF   MOTION. 

weight,  is  that  which  is  meant  by  a  SIMPLE  PENDULUM.  It  is 
evident  that  the  time  of  oscillation  increases  with  the  length 
I  of  the  pendulum. 

IMPULSIVE  FORCE. 

99.  If  any  number  of  different  moving  forces  be  applied 
to  as  many  equal  bodies,  the  velocities  communicated  to 
them  in  the  same  exceedingly  small  interval  of  time,  will  be 
to  one  another  as  the  moving  forces.  For  let  P1?  P2,  repre- 
sent the  moving  forces,  and  f\,  /"„  the  additional  velocities 
they  would  communicate  per  second  if  each  moving  force 
remained  continually  of  the  same  magnitude  (Art.  93.),  then 
would  tf^  tfv  be  the  whole  velocities  communicated  on  this 
supposition  in  t  seconds  ;  let  these  be  represented  by  V,,  Y2 ; 
therefore  by  Art.  94. 

P, :?,::/  :/,::#  :  */,::¥,  :V,. 

The  proposition  is  therefore  true  on  the  supposition  that  Pt 
and  P2  remain  constant  during  the  interval  of  time  t ;  but 
if  t  be  exceedingly  small,  then  whatever  the  pressures  Pt 
and  P2  may  be,  they  may  be  considered  to  remain  the  same 
during  that  time.  Therefore  the  proposition  is  true  generally, 
when,  as  above,  the  moving  forces  are  supposed  to  act  on 
equal  bodies,  or  successively  on  the  same  body,  through 
equal  exceedingly  small  intervals  of  time. 

M  oving  forces  thus  acting  through  exceedingly  small  in- 
tervals of  time  only,  are  called  IMPULSIVE  FORCES. 


THE  PARALLELOGRAM  OF  MOTION. 

100.  If  two  impulsive  forces  P1?  P2,  whose  directions  are  AB 
and  AC.  he  impressed  at  the  same  time  upon 
^°    a  lody  at  A,  which  if  made  to  act  upon  it 
separately  would  cause  it  to  move  through 
AB  and  AC  in  the  same  given  time,  then 
will  the  hody  he  made,  by  the  simultaneous  action  of  these 
impulsive  forces,  to  describe  in  that  time  the  diagonal  AD 
of  the  parallelogram,  of  which  AB  and  AC  are  adjacent 
sides. 

For  the  moving  forces  Pj  and  P3  acting  separately  upon 


INDEPENDENCE  OF  SIMULTANEOUS  MOl'IONS.        87 

the  same  body  through  equal  infinitely  small  times,  commu- 
nicate to  it  velocities  which  are  (Art.  99.)  as  those  forces; 
therefore  the  spaces  AB  and  AC  described  with  these  velo- 
cities in  any  given  time  are  also  as  those  forces.  Since  then 
AB  and  AC  are  to  one  another  as  the  pressures  "Pl  and  Pa, 
therefore  by  the  principle  (Art.  2.)  of  the  parallelogram  of 
pressures,  the  resultant  K  of  Pt  and  P2  is  in  the  direction  of 
the  diagonal  AD,  and  bears  the  same  proportion  to  Px  and 
P2  that  AD  does  to  AB  and  AC. 

Therefore  the  velocity  which  the  resultant  K  of  P,  and  P2 
would  communicate  to  the  body  in  any  exceedingly  small 
time  is  to  the  velocities  which  P,  and  P2  would  separately 
communicate  to  it  in  the  same  time  as  AD  to  AB  and  AC 
(Art.  99.),  and  therefore  the  spaces  which  the  body  would 
describe  uniformly  with  these  three  velocities  in  any  equal 
times  are  in  the  ratio  of  these  three  lines.  But  AB  and  AC 
are  the  spaces  actually  described  in  the  equal  times  by  rea- 
son of  the  impulses  of  Pj  and  P2.  Therefore  AD  is  the  space 
described  in  that  time  by  reason  of  the  impulse  of  R,  that  is, 
by  reason  of  the  simultaneous  impulses  of  Px  and  Pa. 


101.    THE   INDEPENDENCE    OF    SIMULTANEOUS   MOTIONS. 

It  is  evident  that  if  the  body  starting  from  A  had  been 
made  to  describe  AB  in  a  given  time,  and  then 
had  been  made  in  an  equal  time  to  describe 
BD,  it  would  have  arrived  precisely  at  the  same 
point  D  to  which  the  simultaneous  motions 
AC  and  AB  have  brought  it,  so  that  the  body  is  made  to 
move  by  these  simultaneous  motions  precisely  to  the  same 
point  to  which  it  would  have  been  brought  by  those  motions, 
communicated  to  it  successively,  but  in  half  the  time.  The 
following  may  be  taken  as  an  illustration  of  this  principle  of 
the  independence  of  simultaneous  motions.  Let  a  canal-boat 
__  T*  3,  be  imagined  to  extend  across  the  whole 

=  width  of  the  canal,  and  let  it  be  supposed 
that  a  person  standing  on  the  one  bank  at 
A  is  desirous  to  pass  to  a  point  D  on  the 


opposite  bank,  and  that  for  this  purpose,  as  the  boat  passes 
him,  he  steps  into  it,  and  walks  across  it  in  the  direction 
AB,  arriving  at  the  point  B  in  the  boat  precisely  at  the  in- 
stant when  the  motion  of  the  boat  has  carried  it  through 
BD  ;  it  is  clear  that  he  will  be  brought,  by  the  joint  effect 


88  THE   POLYGON   OF  MOTION. 

of  his  own  motion  across  the  boat  and  the  'booths  motion 
along  the  canal,  to  the  point  D  (having  in  reality  described 
the  diagonal  AD),  which  point  he  would  have  reached  in 
double  the  time  if  he  had  walked  across  a  bridge  from  A  to 
B  in  the  same  time  that  it  took  him  to  walk  across  the  boat, 
and  had  then  in  an  equal  time  walked  from  B  to  D  along 
the  opposite  side. 


THE  POLYGON  OF  MOTION. 

102.  Let  any  number  of  impulses  be  communicated  simul- 
taneously to  a  body  at  O,  one  of  which 
would  cause  it  to  move  from  A  to  O  in  a 
given  time,  another  from  B  to  O  in  the 
same  time,  a  third  from  C  to  O  in  that  time, 
and  a  fourth  from  D  to  O.  Complete  the 
parallelogram  of  which  AO  and  BO  are  ad- 
jacent sides  ;  then  the  impulses  AO  and  BO  would  simulta- 
neously cause  the  body  to  move  from  E  to  O  through  the 
diagonal  EO  in  the  time  spoken  of.  Complete  the  parallelo- 
gram EOCF,  and  draw  its  diagonal  OF,  then  would  the  im- 
pulses EO  and  CO,  acting  simultaneously,  cause  the  body  to 
move  through  FO  in  the  given  time  :  but  the  impulse  EO 
produces  the  same  eifect  on  the  body  as  the  impulses  AO 
and  BO  ;  therefore  the  impulses  AO,  BO,  and  CO,  will 
together  cause  the  body  to  move  through  FO  in  the  given 
time.  In  the  same  manner  it  may  be  shown  that  the  im- 
pulses AO,  BO,  CO,  and  DO,  will  together  cause  the  body  to 
move  through  GO  in  a  time  equal  to  that  occupied  by  the 
body's  motion  through  any  one  of  these  lines. 

It  will  be  observed  that  GD  is  the  side  which  completes 
the  polygon  OAEFG,  whose  other  sides  OA,  AE,  EF,  FG, 
are  respectively  equal  and  parallel  to  the  directions 'O  A,  OB, 
OC,  and  OD,  of  the  simultaneous  impulses. 

Instead  of  the  impulses  AO,  &c.  taking  place  simultane- 
ously., if  they  had  been  received  successively,  the  body 
moving  first  from  O  to  A  in  a  given  time ;  then  through 
AE,  which  is  equal  and  parallel  to  OB,  in  an  equal  time ; 
then  through  EF,  which  is  equal  and  parallel  to  OC,  in  that 
time ;  and  lastly  through  FG,  which  is  equal  and  parallel  to 
OD,  in  that  time,  it  would  have  arrived  at  the  same  point  G. 
to  which  these  impulses  have  brought  it  simultaneously,  but 
after  a  period  as  many  times  greater  as  there  are  motions,  so 


THE   PRINCIPLE   OF   D5ALEMBERT.  89 

that  the  principle  of  the  independence  of  simultaneous 
motions  obtains,  however  great  may  be  the  number  of  such 
motions. 

THE  PRINCIPLE  OF  D'ALEMBERT. 

103.  Let  Wj,  W9,  W8,  &c.  represent  the  weights  of  any 
number  of  bodies  in  motion,  and  P15  P2,  P8,  &c.  the  moving 
forces  (Art.  92.)  upon  these  bodies  at  any  given  instant  of 
the  motion,  i.  e.  the  unbalanced  pressures,  or  the  pressures 
which  are  wholly  employed  in  producing  their  motion,  and 
pressures  equal  to  which,  applied  in  opposite  directions, 
would  bring  them  to  rest,  or  to  a  state  of  uniform  motion. 

WWW 

Then   (Art.  95.),   P^-1/,  P.  =  --/.,  P.  =  — ^'Ac. 
y  y  y 

where  f^f^  /,,  &c.  represent  the  additions  of  velocity  which 
the  bodies  would  receive  in  each  second  of  time,  if  the 
moving  force  upon  each  were  to  become,  at  the  instant  at 
which  it  is  measured,  an  uniform  moving  force.  Suppose 
these  bodies,  whose  weights  are  "W15  W2,  W3,  &c.  to  form  a 
system  of  bodies  united  together  by  any  conceivable  mecha- 
nical connection,  on  which  system  are  impressed,  in  any 
way,  certain  forces,  whence  result  the  unbalanced  pressures 
P0  P2,  P3,  &c.  on  the  moving  points  of  the  system.  Now 
conceive  that  to  these  moving  points  of  the  system  there  are 
applied  pressures  respectively  equal  to  Px,  P2,  P3,  &c.  but 
each  in  a  direction  opposite  to  that  in  which  the  motion  of 
the  corresponding  point  is  accelerated  or  retarded.  Then 
will  the  motion  of  each  particular  point  evidently  pass  into 
a  state  of  uniform  motion,  or  of  rest  (Art.  92.).  The  whole 
system  of  bodies  being  thus  then  in  a  state  of  uniform 
motion,  or  of  rest,  the  forces  applied  to  its  different  elements 
must  be  forces  in  equilibrium. 

"Whatever,  therefore,  were  the  forces  originally  impressed 
upon  the  system,  and  causing  its  motion,  they  must,  together 
with  the  pressures  P15  P2,  Ps,  &c.  thus  applied,  produce  a 
state  of  equilibrium  in  the  system ;  so  that  these  forces  (ori- 
ginally impressed  upon  the  system,  and  known  in  Dynamics 
as  the  IMPRESSED  FORCES)  have  to  the  forces  Pj,  P2,  P3,  &c., 
when  applied  in  directions  opposite  to  the  motions  of  their 
several  points  of  application,  the  relation  of  forces  in  equili- 
brium. The  forces  Pn  P2,  P3,  &c.  are  known  in  Dynamics 
as  the  EFFECTIVE  FORCES.  Thus  in  any  system  of  bodies 
mechanically  connected  in  any  way,  so  that  their  motions 


90 

may  mutually  influence  one  another,  if  forces  equal  to  the 
effective  forces  were  applied  in  directions  opposite  to  their 
actual  directions,  these  would  be  in  equilibrium  with  the 
impressed  forces,  which  is  the  principle  of  D'Alembert. 


104.  The  work  accumulated  in\  a  moving  ~body  through  any 
space  is  equal  to  the  worJc  which  must  be  done  upon  it,  in 
an  opposite  direction,  to  overcome  the  effective  force  upon 
it  through  that  space. 

This  is  evident  from  Arts.  68.  and  69.,  since  the  effective 
force  is  the  unbalanced  pressure  upon  the  body. 

If  the  work  of  the  effective  force  be  said  to  be  done  upon 
the  body,*  then  the  work  of  the  effective  force  upon  it  is 
equal  to  the  work  or  power  accumulated  in  it,  and  this  work 
of  the  effective  force  may  be  all  said  to  be  actually  accu- 
mulated in  the  body  as  in  a  reservoir. 


MOTION  OF  TRANSLATION. 

DEFINITION. — When  a  body  moves  forward  in  space,  with- 
out at  the  same  time  revolving,  so  that  all  its. parts  move 
with  the  same  velocity  and  in  parallel  directions,  it  is  said  to 
move  with  a  motion  of  translation  only. 


105.  In  order  that  a  body  may  move  with  a  motion  of  trans- 
lation only,  the  resultant  of  the  forces  impressed  upon  it 
must  have  its  direction  through  the  centre  of  gravity  of 
the  body. 

For  let  Wtf  w^,  ws,  &c.  represent  the  weights  of  the  parts 
or  elements  of  the  body,  and  let  f  represent  the  additional 
velocity  per  second,  which  any  element  receives  or  would 
receive  if  its  motion  were  at  any  instant  to  become  uniformly 
accelerated.  Since  the  motion  is  one  of  translation  only, 
the  value  of  f  is  evidently  the  same  in  respect  to  every 
other  element.  The  effective  forces  Pj,  P2,  P3,  &c.  on  the 
different  elements  of  the  body  are  therefore  represented  by 

^/>7/>7/> &c- &c- 

*  This  cannot  perhaps  be  correctly  said,  since  work  supposes  resistance. 


MOTION    OF    ROTATION.  91 

Now  the  forces  Pa,  P9,  P8,  &c.  are  evidently  parallel  pres- 
sures. Let  X  be  the  distance  of  the  centre  (see  Art.  IT.)  of 
these  parallel  pressures  from  any  given  plane  ;  and  let  a?,,  #3, 
o?8,  &c.  be  the  perpendicular  distances  of  the  elements  w»  wn 
wz,  &c.  that  is,  of  the  points  of  application  of  Pa,  Pa,  P8,  &c. 
from  the  same  plane.  Therefore  (by  equation  18), 


But  this  is  the  expression  (Art.  19.)  for  the  distance  of  the 
centre  of  gravity  from  the  given  plane  ;  and  this  being  true 
of  any  plane,  it  follows  that  the  centre  of  the  parallel  pres- 
sures P15  Pa,  P3,  &c.  which  are  the  effective  forces  of  the 
system,  coincides  with  the  centre  of  gravity  of  the  system, 
and  therefore  that  the  resultant  of  the  effective  forces  passes 
through  the  centre  of  gravity.  Now  the  resultant  of  the 
effective  pressures  must  coincide  in  direction  with  the  result- 
ant of  the  impressed  pressures,  since  the  effective  pressures 
when  applied  in  an  opposite  direction  are  in  equilibrium 
with  the  impressed  pressures  (by  D'Alembert's  principle). 
The  resultant  of  the  impressed  pressures  must  therefore  have 
its  direction  through  the  centre  of  gravity.  Therefore,  &c. 


MOTION  OF  ROTATION  ABOUT  A  FIXED  Axis. 

106.  Let  a  rigid  body  or  system  be  capable  of  motion 
about  the  axis  A.  Let  m1?  ma,  w3,  &c.  represent  the  volumes 
of  elements  of  this  body,  and  ^  the  weight  of  each  unit 
of  volume.  Also  let  /,/,/,  &c.  represent  the  increments 
of  velocity  per  second,  communicated  to  these  elements 
respectively  by  the  action  of  the  forces  impressed  upon  the 
system.  Let  P,,  P2,  P3,  &c.  represent  these  impressed  forces, 
and  Pup*,  &c.  the  perpendicular  distances  from  the  axis  at 
which  they  are  respectively  applied. 

Now  since  pm^  M-m2,  v<m3,  &c.  are  the  weights  of  the  ele- 
ments, and/,/,  &c.  the  increments  of  velocity  they  receive 


92  MOTION  OF   ROTATION. 

per  second,  it  follows  that^/,   f^/,,   ^/,,   &c.  are 

the  effective  forces  upon  them  (Art.  103.).  Let  p,,  pa,  p3,  &c. 
represent  the  distances  of  these  elements  respectively  from 
the  axis  of  revolution,  then  since  their  effective  forces  are 
in  directions  perpendicular  to  these  distances,  the  moments 

of  these  effective  forces  about  the  axis  are  -  —  *-/$»  -  —  -  f^9 

9  '        g 

—  !«/aP«  &c-     Also  PI^,  P^a,  PS^S?  &c.  are  the  moments  of 

g 

the  impressed  forces  of  the  system  about  the  axis.  Now  the 
impressed  forces  P,,  P2,  P3,  &c.,  together  with  the  resistance 
of  the  axis,  which  is  indeed  one  of  the  impressed  forces,  are 
in  equilibrium  with  the  effective  forces  by  D'Alembert's 
principle.  Taking  then  the  axis  as  the  point  from  which  the 
moments  are  measured,  the  sum  of  the  moments  of  P15  P2, 
&c.  must  equal  the  sum  of  the  moments  of  the  effective 
forces,  or 


"Now  let/"  represent  that  value  of  f^fn  &c.  which  corres- 
ponds to  a  distance  unity  from  the  axis.  Since  the  system 
is  rigid,  and  /*,  f^  /„  &c.  represent  arcs  described  about 
it  in  the  same  time  at  the  different  distances  1,  p1?  p2,  &c.  it 
follows  that  these  arcs  are  as  their  distances,  and  therefore 
that/^/p^^/pij/.^/p,,  &c.  Substituting  these  values 
in  tne  preceding  equation,  we  have 


where  I  represents  the  moment  of  inertia  of  the  mass  about 
its  axis  of  revolution.* 

*  If  a  represent  the  angular  velocity,  or  the  velocity  of  an  element  at  dis« 
tance  unity,  then  by  equation  (72),  /=  -f  ^,  .  •  .  a  -J  =  -f  -y  SPpa  ; 


MOTION   OF    ROTATION. 


93 


AG  by  G, 


If  the  impressed  forces  P  be  the  weights  of  the  parts 
of  the  body  and  6  be,  in  any  position  of 
_tf  the  body,  the  inclination  to  the  vertical 
Ay  of  the  line  AG,  drawn  from  A  to  the 
centre  of  gravity  G,  then  since  the  sum  of 
the  moments  of  the  weights  of  the  parts  is 
equal  to  the  moment  of  the  weight  of  the 
whole  mass  collected  in  its  centre  of 
gravity  (Art.  17.),  we  have,  representing 


Mfx  .  G.  sin. 


MG 


therefore  ^equation  I78),f=g— —  sin. 


(79). 


108.  To  find  the  resultant  of  the  effective  forces  on  a  l>ody 
which  revolves  about  a  fixed  axis. 

The  resultant  of  the  effective  forces  upon  a  body  which 
revolves  about  a  fixed  axis,  is  evidently  equal  to  that  single 
force  which  would  just  be  in  equilibrium  with  these  if  there 
were  no  resistance  of  the  axis.  Let  R,  be  that  single  force, 
then  the  moment  of  R  about  any  point  must  equal  the  sum 
of  the  moments  of  the  effective  forces  about  that  point. 

Take  a  point  in  the  axis  for  the  point 
about  which  the  moments  are  measured, 
and  let  L  be  the  perpendicular  distance 
from  A  of  the  resultant  R.  Now,  as  in 
Art.  106.  it  appears  that  the  sum  of  the 
moments  of  the  effective  forces  about  A  is 

ii 
represented  by  f- 


o 
Now  pa  is  the  velocity  of  a  point  at  distance  p,  therefore  "Ppa  is  the  work 

(Art.  50.)  of  the  force  P  per  second ;  therefore  /  Ppadt  is   the   work  of   P 

0 
(equation  40)  in  the  time  t,  which  is  represented  by  U,  therefore  ai*  —  aaa 

«=-f--^y-  which  corresponds  with  the  result  already  obtained.     See  equation 
(51)7   * 


94  MOTION   OF   KOTATION. 

if* 


(80). 

To  determine  the  value  of  E  let  it  be  observed  that  the 
effective  force  -/m^  on  any  particle  ma,  acting  in  a  direc- 


tion  n^m^  perpendicular  to  the  distance  Am,  from  the  axis 
A,  may  be  resolved  into  two  others,  parallel  to  the  two 
rectangular  axes  Ay  and  Aa?,  each  of  which  is  equal  to  the 
product  of  this  effective  force,  whose  direction  is  n,m^  and 
the  cosine  of  the  inclination  of  n,m,  to  the  corresponding 
axis.  'Now  the  inclination  of  mji,  to  Ax  is  the  same  as  the 
inclination  of  Am,  to  Ay,  since  these  two  last  lines  are  per- 
pendicular to  the  two  former.  The  cosine  of  this  inclination 

equals  therefore  -  i  or  ^i,  if  AN.—  y..     Similarly  the  cosine 
Am,       Pl 

of  the  inclination  of  n^m,  to  Ay  equals  -  !  or  ^1  ,  if  AM.l  =  x.. 

Am,      Pl 

The  resolved  parts  in  the  directions  of  Ax  and  Ay  of  the 

effective  force  -  fmj,  are  therefore  -  fm^,  ^   and  -  fm^ 

9  9  Pi  9 

?X  or  -  fm.y,  and  -  fm.x,. 

p,      f  g 

Similarly  the  resolved  parts  in  the  directions  of  Ax  and 
Ay  of  the  effective  force  upon  ma  are  -/^2ys  and  -  fm^ 

ts  t/ 

and  so  of  the  rest. 

The  sums  X  and  Y  of  the  resolved  forces  in  the  directions 
of  Ax  and  Ay  respectively  (Art.  11.)  are  therefore 


.   .  .  =Y; 

g  ^  ff 


and  -  /  {m.x,  +  m^  +  m8a?3  -f  .....  }  =  Y. 

Now  let  Gx  and  G,  represent  the  distances  G2G  and  G:G 
of  the  centre  of  gravity  of  the  body  from  Ay  and  Ax  respec- 
tively, and  let  the  whole  volume  of  the  body  be  represented 
byM, 


MOTION   OF   ROTATION. 

(equation  18),  MG2— m^ 


95 


Now  if  G  be   the  distance  AG  of  the 
centre  of  gravity  from  A,  G=  VGt*  -f  Gaa, 

/.K=-/MG (82). 

y 

Substituting  in  equation  (82)  the  value  off  from  equation 
(78,)  we  have 


And  substituting  in  equation  (80)  for.  R  its  value  from 
equation  (82), 

/-MGL=/-L 

J     g  J    g      > 

I 


MG 


where  L  is  the  distance  of  the  point  of  application  of  the 
resultant  of  the  effective  forces  from  the  axis. 

Now  let  A  be  the  inclination  of  the  resultant  R  to  the 
axis  Aa?, 

/.  (Art.  11.),  R  cos.  d=X,     R  sin.  d=Y, 

Y 

/.tan.  &=x'<>  but  by  equations  (81), 

Y    G 


.-.tan.  4=tan.  AGG0  /J= 

The  inclination  of  the  resultant  R  to  Ax  is  therefore 
equal  to  the  angle  AGGX,  but  the  perpendicular  to  AG  is 
evidently  inclined  to  Ax  at  this  same  angle.  Therefore  the 
direction  of  the  resultant  R  is  perpendicular  to  the  line  AG, 
drawn  from  the  axis  to  the  centre  of  gravity.  Moreover 


96  THE   CENTRE   OF   OSCILLATION. 

its  magnitude  and  the  distance  of  its  point  of  application 
from  A  have  been  before  determined  by  equations  (83) 
and  (84). 

THE  CENTRE  OF  PERCUSSION. 

109.  It  is  evident,  that  if  at  a  point  of  the  body  through 
which  the  resultant  of  the  effective  forces  upon  it  passes, 
there  be  opposed  an  obstacle  to  its  motion,  then  there  will 
be  produced  upon  that  obstacle  the  same  effect  as  though 
the  whole  of  the  effective  forces  were  collected  in  that 
point,  and  made  to  act  there  upon  the  obstacle,  so  that  the 
whole  of  these  forces  will  take  effect  upon  the  obstacle,  and 
there  will  be  no  effect  of  these  forces  produced  else- 
where, and  therefore  no  repercussion  upon  the  axis. 
It  is  for  this  reason  that  the  point  O  in  the  resultant, 
where  it  cuts  the  line  AG  drawn  from  the  axis  to  the 
centre  of  gravity,  is  called  the  CENTRE  OF  PERCUSSION. 
Its  distance  L  from  A  is  determined  by  the  equation 

I-=g- (85), 

which  is  obtained  from  equation  (84)  by  writing  MK2  for  I 
(Art.  80.),  K  being  the  radius  of  gyration.  If  at  the  centre 
of  percussion  the  body  receive  an  impulse  when  at  rest, 
then  since  the  resultant  of  the  effective  forces  thereby  pro- 
duced will  have  its  direction  through  the  point  where  the 
impulse  is  communicated,  it  follows  that  the  whole  impulse 
will  take  effect  in  the  production  of  those  effective  forces, 
and  no  portion  be  expended  on  the  axis. 


THE  CENTRE  OF  OSCILLATION. 

110.  It  has  been  shown  (Art.  98.)  that  in  the  simple  pen- 
dulum, supposed  to  be  a  single  exceedingly  small  element 
of  matter  suspended  by  a  thread  without  weight,  the  time 
of  each  oscillation  is  dependent  upon  the  length  of  this 
thread,  or  the  distance  of  the  suspended  element  from  the 
axis  about  which  it  oscillates.  If  therefore  we  imagine  a 
number  of  such  elements  to  be  thus  suspended  at  different 
distances  from  the  same  axis,  and  if  we  suppose  them,  after 
having  been  at  first  united  into  a  continuous  body,  placed 
in  an  inclined  position,  all  to  be  released  at  once  from  this 


THE   CENTRE   OF   OSCILLATION.  97 

union  with  one  another,  and  allowed  to  oscillate  freely,  it  is 
manifest  that  their  oscillations  will  all  be  performed  in 
different  times.  Now  let  all  these  elements  again  be  con- 
ceived united  in  one  oscillating  mass.  All  being  then  com- 
pelled to  perform  these  oscillations  in  the  same  time,  whilst 
all  tend  to  perform  them  in  different  times,  the  motions  of 
some  are  manifestly  retarded  by  their  connexion  with  the 
rest,  and  those  of  others  accelerated,  the  former  being  those 
which  lie  near  to  the  axis,  and  the  others  those  more  remote ; 
so  that  between  the  two  there  must  be  some  point  in  the 
body  where  the  elements  cease  to  be  retarded  and  begin  to 
be  accelerated,  and  where  therefore  they  are  neither  accele-- 
rated  nor  retarded  by  their  connexion  with  the  rest ;.  an;  elfe-- 
ment  there  performing  its  oscillations  precisely  in>  the  same 
time  as  it  would  do,  if  it  were  not  connected,  with  the-  rest,, 
but  suspended  freely  from  the*  axis  by  at  thread  without: 
weight.  This  point  in  the  body,  at  the  distance  of  whieh 
from  the  axis  a  single  particle,  suspended  freely,  would  per- 
form its  oscillations  precisely  in  the  same  time  that  the  body- 
does,  is  called  the  CENTRE  OF  OSCILLATION. 


The  centre  of    oscillation    coincides  with    the  centre  of 
percussion. 

111.  For  (by  equation  79)  the  increment  of  angular  velo- 
city per  second  f  of  a  body  revolving  about  an  hori- 
zontal axis,  the  forces  impressed  upon  it  being  the 
weights  of  its  parts  only,  is  represented  by  the  for- 

mula ^-y-sin.  d,  where  d  is  the  inclination  to  the  ver- 
tical  of  the   line  AG,  drawn  from  the  axis   to  its 
centre  of  gravity.     But  (by  equation  84),  L=v,  where  L 


is  the  distance  AO  of  the  centre  of  percussion  from  the 
axis, 


/.  fL=g  sin.  6 

Now  it  has  been  shown  (Art.  98.),  that  the  impressed 
moving  force  on  a  particle  whose  weight  is  w,  suspended 
from  a  thread  without  weight,  inclined  to  the  vertical  at  an 
angle  d,  is  represented  by  w  sin.  6  ;  moreover  if/"  represent 

7 


98  THE   CENTRE   OF   OSCILLATION. 

the  increment  of  velocity  per  second  on  this  particle,  then 
—f  is  the  effective  force  upon  it.  Therefore  by  D'Alem- 
bert's  principle, 

7/1 

w  Bin.  «=•/",     :.f=g  sin.  t,     .;  f=fl. 


Now  fL  is  the  increment  of  velocity  at  the  centre  of 
percussion,  andy  is  that  upon  a  single  particle  suspended 
freely  at  any  distance  from  the  axis.  If  such  a  particle 
were  therefore  suspended  at  a  distance  from  the  axis  equal 
to  that  of  the  centre  of  percussion,  since  it  would  receive, 
at  the  same  distance  from  the  axis,  the  same  increments  of 
velocity  per  second  that  the  centre  of  percussion  does,  it 
would  manifestly  move  exactly  as  that  point  does,  and  per- 
form its  oscillations  in  the  same  time  that  the  body  does. 
Therefore,  &c. 


112.  The  centres  of  suspension  and  oscillation  are  reci- 

procal. 

Let  O  represent  the  centre  of  oscillation  of  a  body 
when  suspended  from  the  axis  A  ;  also  let  G  be  its 
centre  of  gravity.  Let  AO=L,  AG—  G,  OG^G,  ; 
also  let  the  radius  of  gyration  about  A  be  repre- 
sented by  K2,  and  that  about  G  by  &2.  Therefore 
(equation  59),  K2=G2+&2; 

Ga  +  1£  &2 

(equation  85),  L=  —  —  =G  +      .....  (8Y), 


Now  let  the  body  be  suspended  from  O  instead  of  A  ; 
when  thus  suspended  it  will  have,  as  before,  a  centre  of 
oscillation.  Let  the  distance  of  this  centre  of  oscillation 
from  O  be  L1? 

* 
.*.  by  equation  (8T),  L^ 


PROJECTILES.  99 


/.  by  equation  (88),  L^^r  +  G^L. 

Since  then  the  centre  of  oscillation  in  this  second  case  is  at 
the  distance  L  from  O,  it  is  in  A';  what  was  before  the 
centre  of  suspension  has  now  therefore  become  the  centre 
of  oscillation.  Thus  when  the  centre  of  oscillation  is  con- 
verted into  the  centre  of  suspension,  the  centre  of  suspen- 
sion is  thereby  converted  into  the  centre  of  oscillation. 
This  is  what  is  meant,  when  it  is  said  that  the  centres  of 
oscillation  and  suspension  are  reciprocal. 


PROJECTILES. 

113.  To  determine  the  path  of  a  body  projected  obliquely 
in  vacuo. 

Suppose  the  whole  time,  T  seconds,  of  the  flight  of  the 

body  to  any  given  point  P 
^--'''"|T  of  its  path,  to  be  divided 
M.-"'''^,.---'?*  into  equal  exceedingly  small 

*ir""^.,--\v'. — i°  intervals,     represented    by 

'"*  "^  "'"' :   -j     AT,  and  conceive  the  whole 

q\JB  effect  of  gravity  upon  the 
projectile  during  each  one 
of  these  intervals  to  be  col- 
lected into  a  single  impulse  at  the  termination  of  that  inter- 
val, so  that  there  may  be  communicated  to  it  at  once,  by 
that  single  impulse,  all  the  additional  velocity  which  is  in 
reality  communicated  to  it  by  gravity  at  the  different  periods 
of  the  small  time  AT. 

Let  AB  be  the  space  which  the  projectile  would  describe, 
with  its  velocity  of  projection  alone,  in  the  first  interval  of 
time  ;  then  will  it  be  projected  from  B  at  the  commence- 
ment of  the  second  interval  of  time  in  the  direction  ABT 
with  a  velocity  which  would  alone  carry  it  through  the  dis- 
tance BK= AB  in  that  interval  of  time  ;  whilst  at  the  same 
time  it  receives  from  the  impulse  of  gravity  a  velocity  such 
as  would  alone  carry  it  vertically  through  a  space  in  that  in- 
terval of  time  which  may  be  represented  by  BF.  By  reason 
of  these  two  impulses  communicated  together,  the  body  will 
therefore  describe  in  the  second  interval  of  time  the  diago- 
nal BC  of  the  parallelogram  of  which  BK  and  BF  are  adja- 


100  PEOJECTILES. 

cent  sides.  At  the  commencement  of  the  third  interval  it 
will  therefore  have  arrived  at  C,  and  will  be  projected  from 
thence  in  the  direction  BOX,  with  a  velocity  which  would 
alone  carry  it  through  CX^BC  in  the  third  interval  ;  whilst 
at  the  same  time  it  receives  an  impulse  from  gravity  com- 
municating to  it  a  velocity  which  would  alone  carry  it 
through  a  distance  represented  by  CG=BF  in  that  interval 
of  time.  These  two  impulses  together  communicate  there- 
fore to  it  a  velocity  which  carries  it  through  CD  in  the  third 
interval,  and  thus  it  is  made  to  describe  all  the  sides  of  the 
polygon  ABCD  ...  P  in  succession.  Draw  the  vertical  PT, 
and  produce  AB,  BC,  CD,  &c.  to  meet  it  in  T,  N,  O  .  .  ., 
and  produce  GC,  HD,  &c.  to  meet  BT  in  K,  L,  &c. 

Now,  since  BC  is  equal  to  CX,  and  CK  is  parallel  to  XL, 
therefore  KL  is  equal  to  BK  or  to  AB. 

Again,  since  CD  is  equal  to  DZ,  and  DL  is  parallel  to  ZM, 
therefore  LM  is  equal  to  KL  or  to  AB  ;  and  so  of  the  rest. 

If  therefore  there  be  n  intervals  of  time  equal  to  AT,  so 
that  there  are  n  sides  AB,  BC,  CD,  &c.  of  the  polygon,  and 
n  divisions  AB,  BK,  &c.  of  the  line  AT,  then  AT,=7iAB  and 


Similarly   CET=(w-2)CX,   therefore  ]Sr6=(^-2)DX= 
(n—  2)BF;  and  so  of  the  remaining  parts  of  TP. 

these  parts  of  TP  are  (n—T)  in  number,  therefore 
-3)W+  ...  \(n—l)  terms}; 


Therefore,  summing  the  series  to  (n—V)  terms. 
TP={2(»-l)-(»-a)}=i    .  BF, 


Now  g  represents  the  additional  velocity  which  gravity 
would  communicate  to  the  projectile  in  each  second,  if  it 
acted  upon  it  alone.  g&T  is  therefore  the  velocity  which  it 
would  communicate  to  it  in  each  interval  of  AT  seconds. 
<?AT  is  therefore  the  velocity  communicated  to  the  body  by 
each  of  the  impulses  which  it  has  been  supposed  to  receive 
from  gravity. 


PROJECTILES.  101 

Now  BF  is  the  space  through  which  it  would  be  carried 
in  the  time  AT  by  this  velocity, 


Also  AT=-, 
n 


Now  this  is  true,  however  small  may  be  the  intervals  of 
time  AT,  and  therefore  if  they  be  infinitely  small,  that  is,  if 
the  impulses  of  gravity  be  supposed  to  follow  one  another  at 
infinitely  small  intervals,  or  if  gravity  be  supposed  to  act,  as 
it  really  does,  continuously. 

But  if  the  intervals  of  time  AT  be  infinitely  small,  then 
the  number  n  of  these  intervals  which  make  up  the  whole 
finite  time  T,  must  be  infinitely  great.  Also  when  n  is  infi- 

nitely great,  -=0. 

In  the  actual  case,  therefore,  of  a  projectile  continually 
deflected  by  gravity,  the  vertical  distance  TP  between  the 
tangent  to  its  path  at  the  point  of  projection,  and  its  position 
P  after  the  flight  has  continued  T  seconds,  is  represented  by 
the  formula 

TP=%gT  .....  (89). 

Moreover  AT^TiAB,  and  AB  is  the  space  which  the  body 
would  describe  uniformly  with  the  velocity  of  projection  in 
the  time  AT,  so  that  ?zAB  is  the  space  which  it  would  de- 
scribe in  the  time  n  .  AT  or  T  with  that  velocity.  If  there- 
fore Y  equal  the  velocity  of  projection,  then 

AT=Y  .  T  .  .  .  .  (90)  ; 

so  that  the  position  of  the  body  after  the  time  T  is  the  same 
as  though  it  had  moved  through  that  time  with  the  velocity 
of  its  projection  alone,  describing  AT,  and  had  then  fallen 
through  the  same  time  by  the  force  of  gravity  alone,  describ- 
ing TP  (see  Art.  101.). 


Let   AM=x,   MP=y,   angle   of 
projection  TAM^a,  velocity  of  projec- 


--*-•£ * 


102  PROJECTILES. 


x  tan.  a—  y=MT—  MF=P=i/r  .....  (91). 
Substituting  the  value  of  T  from  the  preceding  equation, 

.,   a?2  sec.2  a 
tan.  a—   = 


<7  sec.  a        a 

.-.y=0tan.  a~  2y2      •»• 

Let  H  be  the  height  through  which  a  body  must  fall  freely 
by  gravity  to  acquire  the  velocity  Y,  or  the  height  due  to 


that  velocity  ;  then  Y2—  2gTL  (Art.  47.),  therefore  4H=  --  ; 
therefore,  by  substitution, 

(92). 


115.  To  find  the  time  of  the  fiight  of  a  projectile. 

It  has  been  shown  (equation  91),  that  if  T  represent  the 
time  in  seconds  of  the  flight  to  a  point  whose  co-ordinates 
are  x  and  y,  then 

2 
— x  tan.  a— y,     /.  T=  -  \x  tan.  a— y\ , 


(93). 


.•.T=l^»tan.  a—  y  nearly. 


If  the  projectile  descend  again  to  the  horizontal  plane  from 
which  it  was  projected,  and  T  be  the  whole  time  of  its  flight; 
and  X  its  whole  range  upon  the  plane,  then,  since  at  the  ex 
piration  of  the  time  T,  y=Q  and  a?=X, 


-^X  tan.  a=  Jy/Xtan.  a  nearly. 


PROJECTILES.  103 


116.  To  find  the  greatest  horizontal  distance  X,  to  which  a 
projectile  ranges,  having  given  the  elevation  a  and  the 
velocity  V  of  its  projection. 

When  the  projectile  attains  its  greatest  horizontal  range, 
its  height  y  above  the  horizontal  plane 
™  becomes  0,  whilst  the  abscissa  x  of  the 

/  !  point  P,  which  it  has  then  reached  in 

its   path,   becomes    X.       Substituting 
S^s^i?^  these  values  0  and  X,  for  y  and  x  in 

./  \  \       equation  (92),  we  have  0=X  tan.  a— 

*£ JUi, — ::>    X2  sec.2  a 


4H 

.*.  X=4H  tan.  a  cos.2a=4H  sin.  a  cos.  a. 
.•.X=2Hsin.  2« (94). 

If  the  body  be  projected  at  different  angular  elevations, 
but  with  th£  same  velocity,  the  horizontal  range  will  be  the 

greatest  when  sin.  2a  is  the  greatest,  or  when  2a=-,  or  a=j- 


117.  To  find  the  greatest  height  which  a  projectile  will 
attain  in  its  flight  if  projected  with  a  given  velocity  ,  and 
at  a  given  inclination  to  the  horizon. 

Multiplying  both  sides  of  equation 
(92)  by  4H  cos.2  «,  we  have  4H  cos.2  a 
.  y—^R  cos.2  a  tan.  a  .  x—  x?=2T3.  (2 
cos.  a  sin.  a)  x—x*=2TL  sin.  2a  .  x—x*. 
Subtracting  both  sides  of  this  equa- 
2  sm-2  2°S  we  have 


H8  sin2  2a—  4H  cos.2  a  .  y=W  sin.2  2a—  2H  sin.  2a  . 
But  sin.2  2a=4:  sin.3  a  cos.2a, 

/.4H  cos.2  a{H  sin.2  a  -y}  --=  JH  sin.  2a-^8.  .  .  .  (95). 

E"ow  the  second  member  of  this  equation  is  always  a 
positive  quantity,  being  a  square.  The  first  member  is 
therefore  always  positive  ;  therefore  H  sin.2  a—  y  is  always 
positive.  "Whence  it  follows  that  y  can  never  exceed  H 
sin.2  a,  so  that  it  attains  its  greatest  possible  value  when  it 
equals  H  sin.2  a,  a  value  which  it  manifestly  attains  when 


104  PROJECTILES. 

the  first  member  of  the  above  equation  vanishes,  or  when 
a?z=H  sin.  2»,  that  is,  when  x  becomes  equal  to  half  the 
greatest  horizontal  range,  as  is  apparent  from  the  last  pro- 
position; so  that  the  greatest  height  BD  of  the  projectile 
is  represented  by  H  sin.a  a,  a  height  which  it  attains  when 
AD  equals  half  the  horizontal  range. 


118.  The  path  of  a  projectile  in  vacuo  is  a  parabola. 

Let  B  be  the  highest  point  in  the 
flight  of  the  projectile,  and  BD  its 
greatest  height.  Draw  PMt  perpen- 
dicular to  BD.  Let  BM1=a?1,  MJP 


.-.     aj^BD—  M1D=BD—  PM=H  sin.2  a—  y, 

y1=DM=AM—  AD=»—  -  H  sin.  2«. 
Substituting  these  values  in  equation  (95), 

y^-iH  cos.2  «  .  x,  .....  (96), 

which  is  the  equation  to  a  porabola  whose  vertex  is  in 
B,  whose  axis  coincides  with  BD,  and  whose  parameter  is 
4H  cos.2  «. 

The  path  of  a  projectile  in  vacuo  is  therefore  a  parabola, 
whose  vertex  is  at  the  highest  point  attained  by  the  pro- 
jectile, and  whose  axis  is  vertical. 


119.  To  find  the  range  of  a  projectile  upon  an  inclined 

plane. 

Let  ~R  represent  the  range  AP  of  a  projectile  upon  an 
inclined  plane  AB,  whose  inclination  is 
i.     Then  H  and  a  being  taken  to  repre- 
,    sent  the  same  quantities  as  before,  and 
"c  x,  y  being  the  co-ordinates  of  P  to  the 
horizontal  azis  AC,  we  "have 

x=AM=AP  cos.  PAM=K  cos.  i, 
sin.  PAM =R  sin.  ». 


^  Substituting  these  values  of  x  and  y  in  the  general  equa- 
tion (92)  of  the  projectile  we  have 


PROJECTILES.  105 

T,  T-,  Ra  cos.3  i  sec.1  a 

R  sin.  i=R  cos.  i  tan.  a  --  -==  -  . 

Dividing  by  R,  multiplying  by  cos.  «,  and  transposing 

R  cos.8  *  sec.  a 

-  JTJ  --  =cos.  i  sin.  a  —  sm.  »  cos.  a=sm.  (a  —  i), 


(97). 


Now  sin.  (2a  —  «)  —  sin.  «=sin.   {a  +  (a  —  i)|  —  sin.  ja  —  (a  —  . 
i)j  =2  sin.  (a  —  »)  cos.  a. 

Substituting  this  value  of  2  sin.  (a  —  »)  cos.  a  in  the  pre- 
ceding equation,  we  have 


._.  > 

j 


J^ow  it  is  evident  that  if  a  be  made  to  vary,  <  remaining 
the  same,  R  will  attain  its  greatest  value  when  sin.  (2a  —  ») 
is  greatest,  that  is  when  it  equals  unity,  or  when  2a  —  1= 

o,  or  when  a=-  +  -.     This,  then,  is  the  angle  of  elevation 

corresponding  to  the  greatest  range,  with  a  given  velocity 
upon  an  inclined  plane  whose  inclination  is  ». 

If  in  the  preceding  expression  for  the  range  we  substitute 

(  *  I 

)  o~(a~')  f  f°r  a?  tne  value  of  the  expression  will  be  found 

to  remain  the  same  as  it  was  before  ;  for  sin.  (2«—  i)  will,  by 
this  substitution,  become  sin.  jtf—  2(a—  »)—  «}  =sin.  \t—  (2« 
—  1){  =sin.  (2a—  i).  The  value  of  R  remains  therefore  the 

"Tf 

same,  whether  the  angle  of  elevation  be  a  or  s~(a~')- 

And  the  projectile  will  range  the  same  distance  on  the 
plane,  whether  it  be  projected  at  one  of  these  angles  of 
elevation  or  the  other. 

Let  BAG  be  the  inclination  of  the  plane  on  which  the 
projectile  ranges,  and  AT  the  direc- 
tion of  projection.  Take  DAS  equal 
to  BAT.  Then  BAT=TAC-BAC 
=a-i.  And  SAC=:DAC-DAS= 

-—  BAT=~(«—  »)•    The  range  AP 

2  a 

is  therefore  the  same,  whether  TAG  or  SAO  be  the  angle  of 


106  CENTRIFUGAL   FORCE. 

elevation,  and  therefore  whether  AT  or  AS  be  the  direction 
of  projection. 

Draw  AE  bisecting  the  angle  BAD,  then  the  angle  EAC 


The  angle  EAC  is  therefore  that  corresponding  to  the 
greatest  range,  and  AE  is  the  direction  in  which  a  body 
should  be  projected  to  range  the  greatest  distance  on  the 
inclined  plane  AB. 

It  is  evident  that  the  directions  of  projection  AS  and  AT, 
which  correspond  to  equal  ranges,  are  equally  inclined  to 
the  direction  AE  corresponding  to  the  greatest  range. 


120.  The  velocity  of  a  projectile  at  different  points  of  its 
path.  It  has  been  shown  (Art.  56.),  that  if  a  body  move  in 
any  curve  acted  upon  by  gravity,  the  work  accumulated  or 
lost  is  the  same  as  would  be  accumulated  or  lost,  provided 
the  body,  instead  of  moving  in  a  curve,  had  moved  in  the 
direction  of  gravity  through  a  space  equal  to  the  vertical 
projection  of  its  curvilinear  path. 

thus  a  projectile  moving  from  A  to  P  will  accumulate  or 
lose  a  quantity  of  work,  which  is  equal  to  that  which  it  would 
accumulate  or  lose,  had  it  moved  vertically  from  M  to  P,  or 
from  P  to  M,  PM  being  the  projection  of  its  path  on  the 
direction  of  gravity.  5"  ow  the  work  thus  accumulated  or 
lost  equals  one  half  the  difference  between  the  vires  vivce  at 
the  commencement  and  termination  of  the  motion. 

Let  Y  equal  the  velocity  at  A,  and  v  equal  the  velocity  at 

W          "W 

P,  therefore  the  work  —  -J  —  Va—  -J-  —  v*.    Moreover,  the  work 

9  9 

done  through  PM=W  .  PM,    therefore  £—  V—  ^-v*= 


W  .  PM,  therefore  Y2— va=2^MP.     Let  PM=y, 

/.v3— Ya— %gy (99), 

which  determines  the  velocity  at  any  point  of  the  curve. 

CENTRIFUGAL  FORCE. 
121.  Let  a  body  of  small  dimensions  move  in  any  curvi- 


CENTRIFUGAL   FOKCE.  1Q7 

linear  path  AB,  impelled  continnally  towards 
a  given  point  S  (called  a  centre  of  force)  by  a 
given  force,  whose  amount,  when  the  body 
has  reached  the  point  P  in  its  path,  is  repre- 
sented by  F.*  Let  PQ  be  an  exceedingly 
small  portion  of  the  path  of  the  body,  and 
conceive  the  force  F  to  remain  constant  and 
parallel  to  itself,  whilst  this  portion  of  its  path  is  being  de- 
scribed. Then,  if  PR  be  a  tangent  at  P,  and  QR  be  drawn 
parallel  to  SP,  PR  is  the  space  which  the  body  would  have 
traversed  in  the  time  of  describing  PQ,  if  it  had  moved 
with  its  velocity  of  projection  from  P  alone,  and  had  not  been 
attracted  towards  S,  and  RQ  or  PT  (QT  being  drawn  paral- 
lel to  RP)  is  the  space  through  which  it  would  have  fallen 
by  its  attraction  towards  S  alone,  or  if  it  had  not  been  pro- 
jected at  all  from  P.f  Let  v  represent  the  velocity  which 
it  would  have  acquired  on  this  last  supposition,  when  it 
reached  the  point  T.  Therefore  (Art.  66.),  if  w  represent  the 
weight  of  the  body, 


Now  the  velocity  v,  which  the  body  would  have  acquired  in 
falling  through  the  distance  PT  by  the  action  of  the  constant 
1  orce  F,  is  equal  to  double  that  which  would  cause  it  to  de  • 
scribe  the  same  distance  uniformly  in  the  same  time4 

Representing  therefore  by  Y  the  actual  velocity  of  the 
body  in  its  path  at  P,  we  have 


V~PE'  'PE' 

9 

Substituting  this  value  of  v  in  the  preceding  equation, 

*  The  force  here  spoken  of,  and  represented  by  F,  is  the  moving  force,  or 
pressure  on  the  body  (see  Art.  92.),  and  is  therefore  equal  to  that  pressure 
which  would  just  sustain  its  attraction  towards  S. 

f  See  Art.  113.  (equations  89  and  90)  ;  what  is  proved  there  of  a  body  acted 
upon  by  the  force  of  gravity  which  is  constant,  and  whose  direction  is  con- 
stantly parallel  to  itself,  is  evidently  true  of  any  other  constant  force  similarly 
retaining  a  direction  parallel  to  itself.  To  apply  the  same  demonstration  to 
any  such  case,  we  have  only  indeed  to  assume  g  to  represent  another  number 
than  32*. 

1  If  /  represent  the  additional  velocity  per  second  which  F  would  com- 
municate to  the  body,  and  t  the  time  of  describing  PT,  then  (Art.  44.) 


«=/*;  but  (Art.  46.)  PT=^9  =        *=|<;    so  that       is  the   velocity   with 
which  PT  would  be  described  uniformly  in  the  time  t. 


108 


CENTRIFUGAL   FORCE. 


Now  let  a  circle  PQY  be  described  having  a  common  tan 
gent  with  the  curve  AB  in  the  point  P,  and  passing  through 
the  point  Q.  Produce  PS  to  intersect  the  circumference  of 
this  circle  in  Y,  and  join  QY  ;  then  are  the  triangles  PQY 
and  QPR  similar,  for  the  angle  RQP  is  equal  to  the  angle 
QPY  (QE  and  YP  being  parallel),  and  the  angle  QPR  is 
equal  to  the  angle  Q  YP  in  the  alternate  segment  of  the  cir- 


cle.    Therefore         =  therefore  QR=.      Substi- 


tuting this  value   of  QR  in  the   last   equation,  we  have 

QV 


Now  this  is  true,  however  much  PQ  may  be  diminished. 
Let  it  be  infinitely  diminished,  the  supposed  constant  amount 
and  parallel  direction  of  F  will  then  coincide  with  the  actual 
case  of  a  variable  amount  and  inclination  of  that  force,  the 

••    PQ 

ratio  ~~  will  become  a  ratio  of   equality,  and  the  circle 

PQY  will  become  the  circle  of  curvature  at  P,  and  PY  that 
chord  of  the  circle  of  curvature,  which  being  drawn  from  P 
passes  through  S.  Let  this  chord  of  the  circle  of  curvature 
be  represented  by  C, 


The  force  or  pressure  F  thus  determined  is  manifestly 
exactly  equal  to  that  force  by  which  the  body  tends  in  its 
motion  continually  to  fly  from  the  centre  S,  and  may  there- 
fore be  called  its  centrifugal  force.  This  term  is,  however, 
generally  limited  in  its  application  to  the  case  of  a  body  re- 
volving in  a  circle,  and  to  the  force  with  which  it  tends  to 
recede  from  the  centre  of  that  circle  ;  or  if  applied  to  the 
case  of  motion  in  any  other  curve,  then  it  means  the  force 
with  which  the  body  tends  to  recede  from  the  centre  of  the 
circle  of  curvature  to  its  path  at  the  point  through  which  it 
is,  at  any  time,  moving.  When  the  body  revolves  in  a  cir- 
cular path,  the  circle  of  curvature  to  the  path  at  any  one 
point  evidently  coincides  with  it  throughout,  and  the  chord 
of  curvature  becomes  one  of  its  diameters.  Let  the  radius 
of  the  circle  which  the  body  thus  describes  be  represented 
by  R,  then  C=2R  ; 


CENTRIFUGAL   FORCE.  109 


»«>• 


Since  in  whatever  curve  a  body  is  moving,  it  may  be  con- 
ceived at  any  point  of  its  path  to  be  revolving  in  the  circle 
of  curvature  to  the  curve  at  that  point,  the  force  F,  with 
which  it  then  tends  to  recede  from  the  centre  of  the  circle 
of  curvature  is  represented  by  the  above  formula,  B.  being 
taken  to  represent  the  radius  of  curvature  at  the  point  of  its 
path  through  which  it  is  moving. 

If  a  be  the  angular  velocity  of  the  body's  revolution  about 
the  centre  of  its  circle  of  curvature,  then  V=«R  5 

/.F=-a'R  .....  (102). 
9 

122.  From  equation  (100)  we  obtain 


/y 

Now  (Art.  94.)  —  ^  represents  the   additional  velocity  per 

second  f,  which  would  be  communicated  to  a  body  falling 
towards  6,  if  the  body  fell  freely  and  the  force  F  remained 
constant.  Moreover,  by  Art.  47.  it  appears,  that  Y  is  the 
whole  velocity  which  the  body  would  on  this  supposition 
acquire,  whilst  it  fell  through  a  distance  equal  to  JC,  or  to 
one  quarter  of  the  chord  of  curvature.  Thus,  then,  the  velo- 
city of  a  body  revolving  in  any  curve  and  attracted  towards 
a  centre  of  force  is,  at  any  point  of  that  curve,  equal  to  that 
which  it  would  acquire  in  falling  freely  from  that  ppint  to- 
wards the  centre  of  force  through  one  quarter  of  that  chord 
of  curvature  which  passes  through  the  centre  of  force,  if  the 
force  which  acted  upon  it  at  that  point  in  the  curve  re- 
mained constant  during  its  descent.  It  is  in  this  sense  that 
the  velocity  of  a  body  moving  in  any  curve  about  a  centre 
of  force  is  said  to  be  THAT  DUE  TO  ONE  .QUARTER  THE  CHORD 

OF   CURVATRE. 

123.  The  centrifugal  force  of  a  mass  of  finite  dimensions. 

Let  BC  represent  a  thin  lamina  or  slice 
of  such  a  mass  contained  between  two  planes 
exceedingly  near  to  one  another,  and  both 
perpendicular  to  a  given  axis  A,  about 
which  the  mass  is  made  to  revolve. 


110 


CENTRIFUGAL   FORCE. 


Through  A  draw  any  two  rectangular  axes  Ax  and  Ay, 
let  m1  be  any  element  of  the  lamina  whose  weight  is  w^  and 
let  AMX  and  AN^  co-ordinates  of  m1?  be  represented  by  x1 
and  yy  Then  by  equation  (102),  if  a  represent  the  angular 
velocity  of  the  revolution  of  the  body,  the  centrifugal  force 


on  the  element  ml  is  represented  by  —  w^Am^  Let  now  this 

J/ 

force,  whose  direction  is  Am1  be  resolved  into  two  others, 
whose  directions  are  Ax  and  Ay.     The  former  will  be  repre- 


sentedby  —  w^Am^  cos.  xAml7  or  by—  wp^  and  the   latter 

g  2       g 

by  —  wlAml  cos.  yAm1?  or  by—  w$1  ;    and  the    centrifugal 

9  <? 

forces  and  all  the  other  elements  of  the  lamina  being  simi- 
larly resolved,  we  shall  have  obtained  two  sets  of  forces, 
those  of  the  one  set  being  parallel  to  A#,  and  represented 


by  —  w^  —  wjc^  —  WsPsi  &c.  and  those   of  the  other  set 
' 


parallel  to  Ay1  represented  by  —  w^^  __w,,y2,_i03y3,  &c. 

ff  9          9 

Now  if  X  and  Y  represent  the  resolved  parts  parallel  to 
the  directions  of  Ax  and  Ay,  of  the  resultant  of  these  two 
sets  of  forces,  then  (Art.  11.) 


if  G-a  and  G2  represent  the  co-ordinates  AG!  and  AG2  of  the 
centre  of  gravity  G  of  the  lamina,  and  W  its  weight 
(Art.  18.). 

Now  the  whole  centrifugal  force  F  on  the  lamina  is  the 
resultant  of  these  two  sets  of  forces,  and  is  represented  by 

*  (Art.  11.), 


/.  F=  T-W'G/  +     W2G23  =  -  W  VG^  +  eV,  or 


(103), 


where  G  is  taken  to  represent  the  distance  AG  of  the  centre 
of  gravity  of  the  lamina  from  the  axis  of  revolution. 

Moreover,  the  direction  of  this  resultant  centrifugal  force 


CENTRIFUGAL   FORCE. 


IB  through  A,  since  the  direction  of  all  its  components  are 
through  that  point 


124.  From  the  above  formula,  it  is  apparent  that  if  a  body 
revolving  round  a  fixed  axis  be  conceived  to 
be  divided  into  laminae  by  planes  perpendicu- 
lar to  the  axis,  then  the  centrifugal  force  of 
each  such  laminae  is  the  same  as  it  would 
have  been  if  the  whole  of  its  weight  had 
been  collected  in  its  centre  of  gravity;  so 
o!  that  if  the  centres  of  gravity  of  all  the  laminae 

be  in  the  same  plane  passing  through  the 
axis,  then,  since  the  centrifugal  force  on  each  lamina  has  its 
direction  from  the  axis  through  the  centre  of  gravity  of  that 
lamina,  it  follows  that  all  the  centrifugal  forces  of  these 
laminae  are  in  the  same  plane,  and  that  they  are  PARALLEL 
forces,  so  that  their  resultant  is  equal  to  their  sum,  those 
being  taken  with  a  negative  sign  which  correspond  to 
laminae  whose  centres  of  gravity  are  on  the  opposite  side  of 
the  axis  from  the  rest,  and  whose  centrifugal  forces  are 
therefore  in  the  opposite  directions  to  those  of  the  rest. 
Thus  if  F'  represent  the  whole  centrifugal  force  of  such  a 

mass,  then  F'=  —  2WG.     Now  let  W  represent  the  weight 

of  the  whole  mass,  and  G'  the  distance  of  its  centre  of  gra- 
vity from  the  axis,  therefore  sWG=W'G'; 

/.  F'=-WG' (104). 

Li  the  case,  then,  of  a  revolving  body  capable  of  being 
divided  into  laminae  perpendicular  to  the  axis  of  revolution, 
the  centres  of  gravity  of  all  of  which  laminae  are  in  the 
same  plane  passing  through  the  axis,  the  centrifugal  force  is 
the  same  as  it  would  have  been  if  the  whole  weight  of  the 
body  had  been  collected  in  its  centre  of  gravity,  the  same 
property  obtaining  in  this  case  in  respect  to  the  whole  body 
as  obtains  in  respect  to  each  of  its  individual  laminos. 
Since,  moreover,  the  centrifugal  forces  upon  the  laminae  are 
parallel  forces  when  their  centres  of  gravity  are  all  in  the 
same  plane  passing  through  the  axis  of  gravity,  and  since 
their  directions  are  all  in  that  plane,  it  follows  (Art.  16.), 
that  if  we  take  any  point  O  in  the  axis,  and  measure  the 
moments  of  these  parallel  forces  from  that  point,  arid  call 
a?  the  perpendicular  distance  OA  of  any  lamina  BC  from 


112  THE   PRINCIPLE   OF   VIRTUAL    VELOCITIES. 

that  point,  and  H  the  distance  of  their  resultant  from  that 
point,  then 


The  equations  (104:)  and  (105)  determine  the  amount  and 
the  point  of  application  of  the  resultant  of  the  centrifugal 
forces  upon  the  mass,  upon  the  supposition  that  it  can  be 
divided  into  laminae  perpendicular  to  the  axis  of  revolution, 
all  of  which  have  their  centres  of  gravity  in  the  same  plane 
passing  through  the  axis. 

It  is  evident  that  this  condition  is  satisfied,  if  the  body  be 
symmetrical  as  to  a  certain  axis,  and  that  axis  be  in  the 
same  plane  with  the  axis  of  revolution,  and  therefore  if  it 
intersect  or  if  it  be  parallel  to  the  axis  of  revolution. 

If,  in  the  case  we  have  supposed,  2WG  —  O,  that  is,  if  the 
centre  of  gravity  be  in  the  axis  of  revolution,  then  the  cen- 
trifugal force  vanishes.  This  is  evidently  the  case  where  a 
body  revolves  round  its  axis  of  symmetry. 


125.  If  the  centres  of  gravity  of  the  laminae  into  which 
the  body  is  divided  by  planes  perpendicular  to 
the  axis  of  revolution  be  not  in  the  same  plane 
(as  in  the  figure),  then  the  centrifugal  forces  of 
the  different  laminae  will  not  lie  in  the  same 
plane,  but  diverge  from  the  axis  in  different 
directions  round  it.  The  amount  and  direction 
of  their  resultant  cannot  in  this  case  be  deter- 
mined by  the  equations  which  have  been  given 
above. 


THE  PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

126.  If  any  pressure  P,  whose  point  of  application  A  is 
made  to  move  through  the  straight  line  AB,  ~be  resolved 
into  three  others  X,  Y,  Z,  in  the  directions  of  the  three 
rectangular  axes,  Ox,  Oy,  Qz  /  and  if  AC,  AD,  and  AE, 
be  the  projections  of  AB  upon  these  axes,  then  the  work  of 
P  through  AB  is  equal  to  the  sum  of  the  works  of  X,  Y,  Z, 
through  AC,  AD,  and  AE  respectively,  or  X  .  AC-f-Y  . 

E5+Z  .  AE=P  .  AM. 


THE   PRINCIPLE   OF   VIKTtJAL   VELOCITIES.  113 


Let  the  inclinations  of  the  direction 
of  P  to  the  axes  Ox,  Oy,  Oz  respec- 
tively, be  represented  by  a,  ft  y,  and 
the  inclniations  of  AB  to  the  same 
axes  by  an  ft,  yn 


/.  (Art.  12.)  X=P  cos.  a,  Y=P  cos.  ft  Z=P  cos.  y ;  also  AC 

=AB  cos.  a,,  AD=AB  cos.  ft,  AE=rAB  cos.  y15 
/.  X/AC=P.  AB  cos.  a  cos.  «15  Y.  AD^P.  AB  cos.  /3  cos.  ft, 

Z  .  AE=P  .  AB  cos.  y  cos.  y,, 
/.X  .  AC+Y  .  AD  +  Z  .  AE=P  .  AB  {cos.  a  cos.  a^cos.  fr 

cos.  ft  +  cos.  y  cos.  y^. 

But  by  a  well-known  theorem  of  trigonometry,  cos*  «  eos. 
Wj  +  cos.  /3  cos.  ft  +  cos.  y  cos.  7,= cos.  PAB, 

/.X  .  AC+Y  .  AD  +  Z  .  AE=P  .  AB  cos.  PAB; 


but  AB  cos.  PAB= AM ; 

.-.  X  .  AC+Y  .  AD  +  Z  .  AE=P  .  AM. 

But  (Art.  52.)  the  w^ork  of  P  through  AM  is  equal  to  its 
work  through  AB.     Therefore,  &c.* 

127.  If  any  number  of  forces  fie  in  equilibrium  (being  in 
any  way  mechanically  connected  with  one  another),  and  if, 
subject  to  that  connection,  their  different  points  of  appli- 
cation "be  made  to  move,  each  through  any  exceedingly  small 
distance,  then  the  aggregate  of  the  work  of  those  forces, 
whose  points  of  application  are  made  to  move  towards  the 

*  This  proposition  may  readily  be  deduced  from  Art.  53.,  for  pressures  equal 
and  opposite  to  X,  Y,  Z,  would  just  be  in  equilibrium  with  P,  and  these  tend- 
ing to  move  the  point  A  in  one  direction  along  the  line  AB,  P  tends  to  move 
it  in  the  opposite  direction,  therefore  in  the  motion  of  the  point  A  through  AB, 
the  sum  of  the  works  of  X,  Y,  Z,  must  equal  the  work  of  P.     But  the  work  of 
X,  as  its  point  of  application  moves  through  AB,  is  equal  (Art.  52.)  to  the  • 
work  of  X  through  the  projection  of  AB  upon  Aar,  that  is,  through  AC ;  simi-  - 
larly  the  work  of  Y,  as  its  point  of  application  moves  through  AB,  is  equal  to  , 
its  work  through  the  projection  of  AB  upon  Ay,  or  through  AD;  and  so  of  Z. 
The  sum  of  the  works  of  X,  Y,  and  Z,  as  their  point  of  application  is  made  to  • 
move  through  AB,  is  therefore  equal  to  what  would  have  been  the  sum  of  their  - 
works  had  their  points  of  application  been  made  to  move  separately  through  . 
AC,  AD,  AE ;  this  last  sum  is  therefore  equal  to  the  work  of  P  through  AB, 
which  is  equal  to  the  work  of  P  through  AM,  AM  being  the  projection  of  AB  ; 
upon  the  direction  of  P. 

8 


114:  PRINCIPLE    OF    VIRTUAL    VELOCITIES. 

directions  in  which  the  several  forces  applied  to  them  actt 
shall  equal  the  aggregate  of  the  work  of  those  forces,  the 
motions  of  whose  points  of  application  are  opposed  to  the 
directions  of  the  forces  applied  to  them. 

For  let  all  the  forces  composing  such  a  system  be  re- 
solved into  three  sets  of  forces  parallel  to  three  rectangular 
axes,  and  let  these  three  sets  of  parallel  forces  be  repre- 
sented by  A,  B,  and  C  respectively.  Then  must  the  result- 
ant of  the  parallel  forces  of  each  set  equal  nothing.  For  if 
any  of  these  resultants  had  a  finite  value,  then  (by  Art.  12.) 
the  whole  three  sets  of  forces  would  have  a  resultant,  which 
they  cannot,  since  they  are  in  equilibrium. 

Now  let  the  motion  of  the  points  of  application  of  the 
forces  be  conceived  so  small  that  the  amounts  and  directions 
-of  tiie  forces  may  be  made  to  vary,  during  the  motion,  only 
by  aa  exceedingly  small  quantity,  and  so  that  the  resolved 
forces  upon  any  point  of  application  may  remain  sensibly 
unchanged.  Also  let  u^  u^  u^  represent  the  works  of  these 
resolved  feces  respectively  on  any  point,  and  2^  the  sum 
of  all  the  works  of  the  resolved  forces  of  the  set  A,  2u^  the 
;sam  of  all  the  works  of  the  forces  of  the  set  13,  and  2us  of  the 
;set  C.  Now  since  the  parallel  forces  of  the  set  A  have  no 
{resultant,  therefore  (Art.  59.)  the  sum  of  the  works  of  those 
forces  of  this  set,  whose  points  of  application  are  moved 
tom&rds  the  directions  of  their  forces,  is  equal  to  the  sum  of 
the  works  of  those  whose  points  of  application  are  moved 
jfrota>  the  directions  of  their  forces,  so  that  2^=0,  if  the 
values  of  u^  which  compose  this  sum,  be  taken  with  the 
positive  or  negative  sign,  according  to  the  last  mentioned 
condition. 

Similarly  ,  2w2  =  0  and  2^3  =  0,      /.  Zfa  -\~u^  +  ?/3)—  0. 

Now  let  U  represent  the  actual  work  of  that  force  Pn  the 
works  of  whose  components  parallel  to  the  three  axes  are 
represented  by  u^  u^  u3  ;  then  by  the  last  proposition, 


(106); 


in  which  expression  U  is  to  be  taken  positively  or  negatively 
according  to  the  same  condition  as  u^  u»  u3  ;  that  is,  accord- 
ing as  the  work  at  each  point  is  done  in  the  direction  of  the 
corresponding  force,  or  in  a  direction  opposite  to  it.  Hence 
therefore  it  follows,  from  the  above  equations,  that  the  sum 


THE   PRINCIPLE    OF   VIS   YIVA.  115 

of  the  works  in  one  of  these  directions  equals  their  sum  in 
the  opposite  direction.     Therefore,  &c. 

The  projection  of  the  line  described  by  the  point  of  appli- 
cation of  any  force  npon  the  direction  of  that 'force  is  called 
its  VIRTUAL  VELOCITY,  so  that  the  product  of  the  force  by  its 
virtual  velocity  is  in  fact  the  work  of  that  force ;  hence 
therefore,  representing  any  force  of  the  system  by  P,  and 
its  virtual  velocity  by  p,  we  have  Pp=tl,  and  therefore, 
=iO,  which  is  the  principle  of  virtual  velocities.* 


128.  If  there  be  a  system  of  forces  such  that  their  points  of 
application  being  moved  through  certain  consecutive  posi- 
tions, those  forces  are  in  all  such  positions  in  equilibrium, 
then  in  respect  to  any  finite  motion  of  the  points  of  appli- 
cation through  that  series  of  positions,  the  aggregate  of  the 
work  of  those  forces,  which  act  in  the  directions  in  which 
their  several  points  of  application  are  made  to  move,  is  equal 
to  the  aggregate  of  the  work  in  the  opposite  direction. 

This  principle  has  been  proved  in  the  preceding  proposi- 
tion, only  when  the  motions  communicated  to  the  several 
points  of  application  are  exceedingly  small,  so  that  the  work 
done  by  each  force  is  done  only  through  an  exceedingly 
small  space.  It  extends,  however,  to  the  case  in  which  each 
point  of  application  is  made  to  move,  and  the  work  of  each 
force  to  be  done,  through  any  distance,  however  great,  pro- 
vided only  that  in  all  the  different  positions  which  the  points 
of  application  of  the  forces  of  the  system  are  thus  made  to 
take  up,  these  forces  be  in  equilibrium  with  one  another ;  for 
it  is  evident  that  if  there  be  a  series  of  such  positions 
immediately  adjacent  to  one  another,  then  the  principle 
obtains  in  respect  to  each  small  motion  from  one  of  this  set 
of  positions  into  the  adjacent  one,  and  therefore  in  respect 
to  the  sum  of  all  such  small  motions  as  may  take  place  in  the 
system  in  its  passage  from  any  one  position  into  any  other, 
that  is,  in  respect  to  the  whole  motion  of  the  system  through 
the  intervening  series  of  positions.  Therefore,  &c. 

THE  PRINCIPLE  OF  Yis  YIVA. 

129.  If  the  forces  of  any  system  be  not  in  equilibrium  with 
one  another,  then  the  difference  between  the  aggregate  work 

*  This  proof  of  the  principle  of  virtual  velocities  is  given  here  for  the  first 
time. 


116  THE   PKINCTPLE   OF   VIS   VIVA. 

of  those  whose  tendency  is  in  the  direction  of  the  motions 
of  their  several  points  of  application,  and  those  whose  ten- 
dency is  in  the  opposite  direction,  is  equal  to  one  half  the 
aggregate  vis  viva  of  the  system. 

In  each  of  the  consecutive  positions  which  the  bodies  com- 

ring  the  system  are  made  successively  to  take  up,  let  there 
applied  to  each  body  a  force  equal  to  the  effective  force 
(Art.  103.)  upon  that  body,  but  in  an  opposite  direction; 
every  position  will  then  become  one  of  equilibrium. 

Now,  as  the  bodies  which  compose  the  system  and  the 
various  points  of  application  of  the  impressed  forces  move 
through  any  finite  distances  from  one  position  into  another, 
let  2^  represent  the  aggregate  work  of  those  impressed 
forces  whose  directions  are  towards  the  directions  of  the 
motions  of  their  several  points  of  application,  and  let  2^2 
represent  the  work  of  those  impressed  forces  which  act  in  the 
opposite  directions  ;  also  let  2u3  represent  the  aggregate 
work  of  forces  applied  to  the  system  equal  and  opposite  to 
the  effective  forces  upon  it  ;  the  directions  of  these  forces 
opposite  to  the  effective  forces  are  manifestly  opposite  also 
to  the  directions  of  the  motions  of  their  several  points  of 
application,  so  that  on  the  whole  2^  is  the  aggregate  work 
of  those  forces  whose  directions  are  towards  the  motions  of 
their  several  points  of  application,  and  2wa  +  5to8  the  aggre- 
gate work  opposed  to  them.  Since  therefore,  by  D'Alem- 
berf  s  principle,  an  equilibrium  obtains  in  every  consecutive 
position  of  the  system,  it  follows  by  the  last  proposition, 
that 


(107). 


Now  us  is  taken  to  represent  the  work  of  a  force  equal  and 
opposite  to  the  effective  force  upon  any  body  of  the  system  ; 
but  the  work  of  such  a  force  through  any  space  is  equal  to 
the  work  which  the  effective  force  (being  unopposed)  accu- 
mulates in  the  body  through  that  space  (Art.  69.),  or  it  is 
equal  to  one  half  the  difference  of  the  vires  vivse  of  the  body 
at  the  commencement  and  termination  of  the  time  during 
which  that  space  is  described  (Art.  67.).  Therefore  2^3 
equals  one  half  the  aggregate  difference  of  the  vires  vivce  of 
the  system  at  the  two  periods  ; 

-O  .....  (108). 


POSITION   OF   MAXIMUM   OR   MINIMUM   VIS   VIVA.  117" 

Thus  then  it  follows,  that  the  difference  between  the  aggre 
gate  work  2  u,  of  those  forces,  the  tendency  of  each  of  which 
is  towards  the  direction  of  the  motion  of  its  point  of  applica- 
tion, and  that  2w2  of  those  the  direction  of  each  of  which  is 
opposed  to  the  motion  of  its  point  of  application  (or,  in  other 
words  the  difference  between  the  aggregate  work  of  the 
accelerating  forces  of  the  system  and  that  of  the  retarding 
forces),  is  equal  to  one  half  the  vis  viva  accumulated  or  lost 
in  the  system  whilst  the  work  is  being  done,  which  is  the 
PRINCIPLE  OF  Yis  VIVA. 


130.  One  half  the  vis  viva  of  the  system  measures  its 
accumulated  work ;  the  principle  of  vis  viva  amounts, 
therefore,  to  no  more  than  this,  that  the  entire  difference 
between  the  work  done  by  those  forces  which  tend  to  accele- 
rate the  motions  of  the  parts  of  the  system  to  which  they 
are  applied,  and  those  which  tend  to  retard  them,  is  accu- 
mulated in  the  moving  parts  of  the  system,  no  work 
whatever  being  lost,  but  all  that  accumulated  which  is  done 
upon  it  by  the  forces  which*  tend  to  accelerate  its  motion, 
above  that  which  is  expended  upon  the  retarding  forces. 

This  principle  has  been  proved  generally  of  any  mechani- 
cal system ;  it  is  therefore  true  of  the  most  complicate  1 
machin^.  The  entire  amount  of  work  done  by  the  moving 
power,  whatever  it  may  be,  upon  that  machine,  is  yielde  I 
partly  at  its  working  points  in  overcoming  the  resistancos 
opposed  there  to  its  motion  (that  is,  in  doing  its  useful 
work),  it  is  partly  expended  in  overcoming  the  friction  and 
other  prejudicial  resistances  opposed  to  the  motion  of  the 
machine  between  its  moving  and  its  working  points,  and  all 
the  rest  is  accumulated  in  the  moving  parts  of  the  machine, 
ready  to  be  yielded  up  under  any  deficiency  of  the  moving 
power,  or  to  carry  on  the  machine  for  a  time,  should  the 
operation  of  that  power  be  withdrawn. 


131.  When  the  forces  of  any  system  (not  in  equilibrium  in 
every  position  which  the  parts  of  that  system^  may  be 
made  to  take  up]  pass  through  a  position  of  equilibrium, 
the  vis  viva  of  the  system  'becomes  a  maximum  or  a 
minimum. 

For,  as  in  Art.  129.,  let  the  aggregate  work  done  in  the 
directions  of  the  motions  of  the  several  parts  of  the  system 


US  POSITION   OF   MAXIMUM   OK   MINIMUM   VIS    VIVA. 

be  represented  by  2^15  and  the  aggregate  work  done  in 
directions  opposed  to  the  motions  of  the  several  parts  by 
2u^  then  (Art.  129.),  one  half  the  acquired  vis  viva  of 
system =2^—2^.  E~ow  as  the  system  passes  from  any  one 
position  to  any  other,  each  of  the  quantities  2^  and  2?^a  is 
manifestly  increased.  If  2^  increases  by  a  greater  quan- 
tity than  2^2,  then  the  ms  viva  is  increased  in  this  change 
of  position ;  if,  on  the  contrary,  it  is  increased  by  a  less 
quantity  than  2^2,  then  the  vis  viva  is  diminished.  Thus  if 
A2wl  and  A2?^2  represent  the  increments  of  2^  and  2^a  in 
this  change  of  position,  then  (2^1  +  A2^1)— (2u9  +  A2u9),  or 
(2^x — 2^2)  -j-  (A2u1 — A2t£2)5  representing  one  half  the  vis 
viva  after  this  change  of  position,  it  is  manifest  that  the  vis 
viva  is  increased  or  diminished  by  the  change  according  a& 
A2^  is  greater  or  less  than  A2^2 ;  and  that  if  AI;^  be  equa) 
to  A2^2  then  no  change  takes  place  in  the  amount  of  the 
vis  viva  of  the  system  as  it  passes  from  the  one  position  to 
the  other. 

Now  from  the  principle  of  virtual  velocities  (Art.  127.), 
it  appears,  that  precisely  this  case  occurs  as  the  system 
passes  through  a  position  of  equilibrium,  the  aggregate 
work  of  those  forces  whose  tendency  is  to  accelerate  the 
motions  of  their  points  of  application  then  precisely  equal- 
ling that  of  the  forces  whose  tendency  is  opposed  to  these 
motions.  For  an  exceeding  small  change  of  position  there- 
fore, passing  through  a  position  of  equilibrium,  A2^=A2^3J 
an  equality  which  does  not,  on  the  other  hand,  obtain, 
unless  the  body  do  thus  pass  through  a  position  of  equili- 
brium. 

Since  then  the  sum  2^ — 2^25  and  therefore  the  aggregate 
vis  viva  of  the  system,  continually  increases  or  diminishes 
up  to  a  position  of  equilibrium,  and  then  ceases  (for  a  cer- 
tain finite  space  at  least)  to  increase  or  diminish,  it  follows, 
that  it  is  in  that  position  a  maximum  or  a  minimum. 
Therefore,  &c. 


132.  When  the  forces  of  any  system  pass  through  a  position 
of  equilibrium,  the  vis  viva  "becomes  a  maximum  or  a 
minimum,  according  as  that  position  is  one  of  stable  or 
unstable  equilibrium. 

For  it  is  clear  that  if  the  vis  viva  be  a  maximum  in  any 
position  of  the  equilibrium  of  the  system,  so  that  after  it 
Has  passed  out  of  that  position  into  another  at  some  finite 


STABLE   AND   UNSTABLE   EQUILIBRIUM. 

distance  from  it,  the  acquired  vis  viva  may  have  become 
less  than  it  was  before,  then  the  aggregate  work  of  the 
forces  which  tend  to  accelerate  the  ^notion  between  these 
two  positions  must  have  been  less  than  that  of  the  forces 
which  tend  to  retard  the  motion  (Art.  131.).  Now  suppose 
the  body  to  have  been  placed  at  rest  in  this  position  of 
equilibrium,  and  a  small  impulse  to  have  been  communi- 
cated to  it,  whence  has  resulted  an  aggregate  amount  of 
vis  viva  represented  by  2mY2.  In  the  transition  from  the 
first  to  the  second  position,  let  this  vis  viva  have  become 
2mv*  ;  also  let  the  aggregate  work  of  the  forces  which  have 
tended  to  accelerate  the  motion,  between  the  two  positions, 
be  represented  by  2U,,  and  that  of  the  forces  which  have 
tended  to  retard  the  motion  by  2U2  ;  then,  for  the  reasons 
assigned  above,  it  appears  that  2U?  is  greater  than  21^. 
Moreover,  by  the  principle  of  vis  viva, 


in  which  equation  the  quantity  2(2  TJ2  —  2U,)  is  essentially 
positive,  in  respect  to  the  particular  range  of  positions 
through  which  the  disturbance  is  supposed  to  take  place.* 

For  every  one  of  these  positions  there  must  then  be  a 
certain.  value  of  2mV2,  that  is,  a  certain  original  impulse 
and  disturbance  of  the  system  from  its  position  of  equili- 
brium, which  will  cause  the  second  member  of  the  above 
equation,  and  therefore  its  first  member  2mv*,  to  vanish. 
Now  every  term  of  the  sum  ^mvz  is  essentially  positive  ; 
this  sum  cannot  therefore  vanish  unless  each  term  of  it 
vanish,  that  is,  unless  the  velocity  of  each  body  of  the 
system  vanishes,  or  the  whole  be  brought  to-  rest.  This 
repose  of  the  system  can,  however,  only  be  instantaneous  ; 
for,  by  supposition,  the  position  into  which  it  has  been  dis- 
placed is  not  one  of  equilibrium.  Moreover,  the  displace- 
ment of  the  system  cannot  be  continued  in  the  direction  in 
which  it  has  hitherto  taken  place,  for  the  negative  term  in 
the  second  member  of  the  above  equation  would  yet  fkrther 
be  increased  so  as  to  exceed  the  positive  term,  and  the  first 

*  The  disturbance  is  of  course  to  be  limited  to  that  particular  range  of 
positions  to  which  the  supposed  position  of  equilibrium  stands  in  the  relation 
of  a  position  of  maximum  vis  viva.  If  there  be  other  positions  of  equili- 
brium of  the  system,  there  will  be  other  ranges  of  adjacent  positions,  in 
respect  to  each  of  which  there  obtains  a  similar  relation  of  maximum  or  mini- 
mum vis  viva. 


i20  STABLE   AND   UNSTABLE   EQUILIBRIUM. 

member  2mv*  would  thus  become  negative,  which  ia 
impossible. 

The  motion  of  the  system  can  then  only  be  continued  by 
the  directions  of  the  motions  of  certain  of  the  elements 
which  compose  it  being  changed,  so  that  the  corresponding 
quantities  by  which  SU,  and  2U2  are  respectively  increased 
may  change  their  signs,  and  the  whole  quantity  ^Uj  —  2U2 
which  before  increased  continually  may  now  continually 
diminish.  This  being  the  case,  2mv*  will  increase  again 
until,  when  st^  —  2U2=0,  it  becomes  again  equal  to  %m Vs ; 
that  is,  until  the  system  acquires  again  the  vis  viva  with 
which  its  disturbance  commenced. 

Thus,  then,  it  has  been  shown,  that  in  respect  to  every 
one  of  the  supposed  positions  of  the  system*  there  is  a  cer- 
tain impulse  or  amount  of  vis  viva,  which  being  communi- 
cated to  the  system  when  in  equilibrium,  will  just  cause  it 
to  oscillate  as  far  as  that  position,  remain  for  an  instant  at 
rest  in  it,  then  return  again  towards  its  position  of  equili- 
brium, and  re-acquire  the  vis  viva  with  which  its  displace- 
ment commenced.  Now  this  being  true  of  every  position 
of  the  supposed  range  of  positions,  it  follows  that  it  is  true 
of  every  disturbance  or  impulse  which  will  not  carry  the 
system  beyond  this  supposed  range  of  positions ;  so  that, 
having  been  displaced  by  any  such  disturbance  or  impulse, 
the  system  will  constantly  return  again  towards  the  position 
of  equilibrium  from  which  it  set  out,  and  is  STABLE  in 
respect  to  that  position. 

On  the  other  hand,  if  the  supposed  position  of  equili- 
brium be  one  in  which  the  vis  viva  is  a  minimum,  then  the 
aggregate  work  of  the  forces  which  tend  to  accelerate  the 
motion  must,  after  the  system  has  passed  through  that  posi- 
tion, exceed  that  of  the  forces  which  tend  to  retard  the 
motion ;  so  that,  adopting  the  same  notation  as  before,  2U, 
must  be  greater  than  2U25  and  the  second  member  of  the 
equation  essentially  positive.  Whatever  may  have  been  the 
original  impulse,  and  the  communicated  vis  viva  2mY2, 
Sm^2  must  therefore  continually  increase ;  so  that  the  whole 
system  can  never  come  to  a  position  of  instantaneous  repose  ;f 
but  on  the  contrary,  the  motions  of  its  parts  must  continu- 
ously increase,  and  it  must  deviate  continually  farther  from 
its  position  of  equilibrium,  in  which  position  it  can  never 

*  That  is,  of  that  range  of  positions  over  which  the  supposed  position  of 
equilibrium  holds  the  relation  of  a  position  of  maximum  vis  viva. 

f  Within  that  range  of  positions  over  which  the  supposed  position  of 
equilibrium  holds  the  relation  of  minimum  vis  viva. 


DYNAMICAL   'STABILITY.  121 

rest.      The  position  is  thus   one  of  unstable  equilibrium 
Therefore,  &c. 

DYNAMICAL  STABILITY.* 

If  a  body  be  made,  by  the  action  of  certain  disturbing 
forces,  to  pass  from  one  position  of  equilibrium  into  another, 
and  if  in  each  of  the  intermediate  positions  these  forces  are 
in  excess  of  the  forces  opposed  to  its  motion,  it  is  obvious 
that,  by  reason  of  this  excess,  the  motion  will  be  continually 
accelerated,  and  that  the  body  will  reach  its  second  position 
with  a  certain  finite  velocity,  whose  eifect  (measured  under 
the  form  of  vis  viva)  will  be  to  carry  it  beyond  that  position. 
This  however  passed,  the  case  will  be  reversed,  the  resist- 
ances will  be  in  excess  of  the  moving  forces,  and  the  body's 
velocity  being  continually  diminished  and  eventually  de- 
stroyed, it  will,  after  resting  for  an  instant,  again  return 
towards  the  position  of  equilibrium  through  which  it  had 
passed.  It  will  not  however  finally  rest  in  this  position  until 
it  has  completed  other  oscillations  about  it.  Now  the  am- 
plitude of  the  first  oscillation  of  the  body  beyond  the  posi- 
tion in  which  it  is  finally  to  rest,  being  its  greatest  ampli- 
tude of  oscillation,  involves  practically  an  important  condi- 
tion of  its  stability ;  for  it  may  be  an  amplitude  sufficient  to 
carry  the  body  into  its  next  adjacent  position  of  equilibrium, 
which  being,  of  necessity,  a  position  of  unstable  equilibrium, 
the  motion  will  be  yet  further  continued  and  the  body 
overturned.  Different  bodies  requiring  moreover  different 
amounts  of  work  to  be  done  upon  them  to  produce  in  all  the 
same  amplitude  of  oscillation,  that  is  (relatively  to  that  am- 
plitude) the  most  stable  which  requires  the  greatest  amount 
of  work  to  be  so  done  upon  it.  It  is  this  condition  of  stabi- 
lity, dependent  upon  dynamical  considerations,  to  which,  in 
the  following  paper,  the  name  of  dynamical,  stability  is 
given.  * 

I  cannot  find  that  the  question  has  before  been  considered 
in  this  point  of  view,  but  only  in  that  which  determines 
whether  any  given  position  be  one  of  stable,  unstable,  or 
mixed  equilibrium  ;  or  which  determines  what  pressure  is 
necessary  to  retain  the  body  at  any  given  inclination  from 
such  a  position. 

*  Extracted  from  a  paper  "  On  Dynamical  Stability,  and  on  the  Oscillations 
of  Floating  Bodies,"  by  the  author  of  this  work,  published  in  the  Transactions 
of  the  Royal  Society,  Part.  II.  for  1850.  The  remainder  of  the  paper  will  be 
found  in  the  Appendix. 


122  DYNAMICAL    STABILITY. 

1.  To  the  discussion  of  the  conditions  of  the  dynamicaj 
stability  of  a  body  the  principle  of  vis  viva  readily  lends 
itself.  That  principle,*  when  translated  into  a  language 
which  the  labours  of  M.  PONCELET  have  made  familiar  to 
the  uses  of  practical  science,  may  be  stated  as  follows  :• — 

"  When,  being  acted  upon  by  given  forces,  a  body  or  sys- 
tem of  bodies  has  been  moved  from  a  state  of  rest,  the  differ- 
ence between  the  aggregate  work  of  those  forces  whose 
tendencies  are  in  the  directions  in  which  their  points  of 
application  have  been  moved,  and  that  of  the  forces  whose 
tendencies  are  in  the  opposite  direction,  is  equal  to  one-half 
the  vis  viva  of  the  system." 

Thus,  if  2^  be  taken  to  represent  the  aggregate  work  of 
the  forces  by  which  a  body  has  been  displaced  from  a  posi- 
tion in  which  it  was  at  rest,  and  2^Q  the  aggregate  work 
(during  this  displacement)  of  the  other  forces  applied  to  it ; 
and  if  the  terms  which  compose  2^  and  2^2  be  understood 
to  be  taken  positively  or  negatively,  according  as  the  ten- 
dencies of  the  corresponding  forces  are  in  the  directions  in 
which  their  points  of  application  have  been  made  to  move 
or  in  the  opposite  directions ;  then  representing  the  aggre- 
gate vis  viva  of  the  body  by  -  2wv*. 

t/ 

2^  +  2^  =^2^', (!'). 

Now  2i£2  representing  the  aggregate  work  of  those  forces 
which  acted  upon  the  body  in  the  position  from  which  it  has 
been  moved,  may  be  supposed  to  the  known ;  2^  may  there- 
fore be  determined  in  terms  of  the  vis  viva,  or  conversely. 

2.  In  the  extreme  position  into  which  the  body  is  made  to 
oscillate  and  from  which  it  begins  to  return,  it,  for  an  instant, 
rests.  In  this  position,  therefore,  its  vis  viva  disappears,  and 
we  have 

2^+2^=0 (2'). 

This  equation,  in  which  2-^  and  2^2  are  functions  of  the 
impressed  forces  and  of  the  inclination,  determines  the  ex- 
treme position  into  which  the  body  is  made  to  roll  by  the 
action  of  given  disturbing  forces ;  or,  conversely,  it  deter- 
mines the  forces  by  which  it  may  be  made  to  roll  into  a 
given  extreme  position. 

*  See  Art.  129. 


DYNAMICAL    STABILITY.  123 

3.  The  position  in.  which  it  will  finally  rest  is  determined 
by  the  maximum  value  of  2^  +  2^  in  equation  (I/)  ;  for,  by 
a  well-known  property,  the  vis  viva  of  a  system*  attains  a 
maximum  value  when  it  passes  through  a  position  of  stable, 
and  a  minimum,  when  it  passes  through  a  position  of  unstable 
equilibrium.      The  extreme  position  into  which  the  body 
oscillates  is  therefore  essentially  different  from  that  in  which 
it  will  finally  rest. 

4.  Different  bodies,  requiring  different  amounts  of  work  to 
be  done  upon  them  to  bring  them  to  the  same  given  inclina- 
tion, that  is  (relatively  to  that  inclination)  the  most  stable 
which  requires  the  greatest  amount  of  work  to  be  so  done 
upon  it,  or  in  respect  to  which  ^u{  is  the  greatest.     If,  in- 
stead of  all  being,  brought  to  the  same  given  inclination,  each 
is  brought  into  a  position  of  unstable  equilibrium,  the  corre- 
sponding value  of  2^  represents  the  amount  of  work  which 
must  be  done  upon  it  to  overthrow  it,  and  may  be  considered 
to  measure  its  absolute,  as  the  former  value  measures  its 
relative  dynamical  stability,  f     The  absolute  dynamical  sta- 
bility of  a  body  thus  measured  I  propose  to  represent  by  the 
symbol  U,  and  its  relative  dynamical  stability,  as  to  the 
inclination  0,  by  U(0). 

The  measure  of  the  absolute  dynamical  stability  of  a  body 
is  the  maximum  value  of  its  relative  stability,  or  U  the  max- 
imum of  U(d)  ;  for  whilst  the  body  is  made  to  incline  from 
its  position  of  stable  equilibrium,  it  continually  tends  to 
return  to  it  until  it  passes  through  a  position  of  unstable 
equilibrium,  when  it  tends  to  recede  from  it  ;  the  aggregate 
amount  of  work  necessary  to  produce  this  inclination  must 
therefore  continually  increase  until  it  passes  through  that 
position  and  afterwards  diminish. 

5.  The  work  opposed  by  the  weight  of  a  body  to  any 
change  in  its  position  is  measured  by  the  product  of  the 
vertical  elevation  of  its  centre  of  gravity  by  its  weight.;): 
Kepresenting  therefore  by  W  the  weight  of  the  body,  and 
by  AH  the  vertical  displacement  of  its  centre  of  gravity 
when  it  is  made  to  incline  through  an  angle  0,  and  observ- 
ing that  the  displacement  of  this  point  is  in  a  direction  oppo- 
site to  that  in  which  the  force  applied  to  it  acts,  we  have 

,  and  by  equation  (2'), 


*  Art.  132. 

f  It  is  obvious  that  the  absolute  dynamical  stability  of  a  body  may  be 
greater  than  that  of  another,  whilst  its  stability,  relatively  to  a  given  inclina- 
tion,  is  less  ;  less  work  being  required  to  incline  it  than  the  other  at  that 
angle,  but  more,  entirely  to  overthrow  it. 

f  Art.  60. 


FKICTION. 


(3). 


If  therefore  no  other  force  than  its  weight  be  opposed  to  a 
body's  being  overthrown,  its  absolute  dynamical  stability, 
when  resting  on  a  rigid  surface,  is  measured  by  the  product 
of  its  weight  hy  the  height  through  which  its  centre  of  gravity 
must  he  raised  to  bring  it  from  a  stable  into  an  unstable 
position  of  equilibrium. 

6.  The  Dynamical  Stability  of  Floating  Bodies.  —  The 
action  of  gusts  of  wind  upon  a  ship,  or  of  blows  of  the  sea, 
being  measured  in  their  'eifects  upon  it  by  their  work,  that 
vessel  is  the  most  stable  under  the  influence  of  these,  or  will 
roll  and  pitch  the  least'  (other  things  being  the  same),  which 
requires  the  greatest  amount  of  work  to  be  done  upon  it  to 
bring  it  to  a  given  inclination  ;  or,  in  respect  to  which  the 
relative  dynamical  stability  U  (4)  is  the  greatest  for  a  given 
value  of  0.  In  another  sense,  that  ship  may  be  said  to  be  the 
most  stable  which  would  require  the  greatest  amount  of  work 
to  be  done  upon  it  to  bring  it  into  a  position  from  which  it 
would  not  again  right  itself,  or  whose  absolute  dynamical 
stability  U  is  the  greatest.  Subject  to  the  one  condition, 
the  ship  will  roll  the  least,  and  subject  to  the  other,  it  will 
be  the  least  likely  to  roll  over. 

Thus  the  theory  of  dynamical  stability  involves  a  question 
of  naval  construction.  It  will  be  found  discussed  in  its  ap- 
plication to  this  question  in  the  Appendix. 


FBICTIOK 

133.  It  is  a  matter  of  constant  experience,  that  a  certain 
resistance  is  opposed  to  the  motion  of  one  body  on  the  sur- 
face of  another  under  any  pressure,  however  smooth  may  be 
the  surfaces  of  contact,  not  only  at  the  first  commencement, 
but  at  every  subsequent  period  of  the  motion  ;  so  that,  not 
only  is  the  exertion  of  a  certain  force  necessary  to  cause  the 
one  body  to  pass  at  first  from  a  state  of  rest  to  a  state  of  mo- 
tion upon  the  surface  of  the  other,  but  that  a  certain  force  is 
further  requisite  to  keep  up  this  state  of  motion.  The  resist- 
ance thus  opposed  to  the  motion  of  one  body  on  the  surface 
of  another  when  the  two  are  pressed  together,  is  called  fric- 


FRICTION.  125 

tion ;  that  which  opposes  itself  to  the  transition  from  a  state 
of  continued  rest  to  a  state  of  motion  is  called  the  friction 
of  quiescence  •  that  which  continually  accompanies  the  state 
of  motion  is  called  the  friction  of  motion. 

The  principal  experiments  on  friction  have  been  made  by 
Coulomb*,  Vince,  G.  Kennief,  K  "Wood;):,  and  recently 
(at  the  expense  of  the  French  Government)  by  Morin.§ 
They  have  reference,  first,  to  the  relation  of  the  friction 
of  quiescence  to  the  friction  of  motion  ;  secondly,  to  the 
variation  of  the  friction  of  the  same  surfaces  of  contact  under 
different  pressures  /  thirdly,  to  the  relation  of  the  friction  to 
the  extent  of  the  surface  of  contact ;  fourthly,  to  the  relation 
of  the  amount  of  the  friction  of  motion  to  the  velocity  of  the 
motion  ;  fifthly,  to  the  influence  of  unguents  on  the  laws  of 
friction,  and  on  its  amount  under  the  same  circumstances  of 
pressure  and  contact.  The  following  are  the  principal  facts 
which  have  resulted  from  these  experiments ;  they  consti- 
tute the  laws  of  friction. 

1st.  That  the  friction  of  motion  is  subject  to  the  same 
laws  with  the  friction  of  quiescence  (about  to  be  stated),  but 
agrees  with  them  more  accurately.  That,  under  the  same 
circumstances  of  pressure  and  contact,  it  is  nevertheless  dif- 
ferent in  amount. 

2ndly.  That,  when  no  unguent  is  interposed,  the  friction 
of  any  two  surfaces  (whether  of  quiescence  or  of  motion)  is 
directly  proportional  to  the  force  with  which  they  are  pressed 
perpendicularly  together  (up  to  a  certain  limit  of  that  pres- 
sure per  square  inch),  so  that,  for  any  two  given  surfaces 
of  contact,  there  is  a  constant  ratio  of  the  friction  to  the  per- 
pendicular pressure  of  the  one  surface  upon  the  other, 
Whilst  this  ratio  is  thus  the  same  for  the  same  surfaces  of 
contact,  it  is  different  for  different  surfaces  of  contact.  The 
particular  value  of  it  in  respect  to  any  two  given  surfaces 
of  contact  is  called  the  CO-EFFICIENT  of  friction  in  re- 
spect to  those  surfaces.  The  co-efficients  of  friction  in  respect 
to  those  surfaces  of  contact,  which  for  the  most  part  form  the 
moving  surfaces  in  machinery,  are  collected  in  a  table,  which 
will  be  found  at  the  termination  of  Art.  140. 

3rdly.  That,  when  no  unguent  is  interposed,  the  amount 
of  the  friction  is,  in  every  case,  wholly  independent  of  the 
extent  of  the  surfaces  of  contact,  so  that  the  force  with  which 
two  surfaces  are  pressed  together  being  the  same,  and 

Mem.  des  Sav.  Etrang.  1781.  t  Phil-  Trans-  1829. 

A  Practical  Treatise  on  Rail-roads,  3d  ed.  chap.  76. 
Mem.  de  1'Institut.  1833,  1834,  1838. 


126  FRICTION. 

not  exceeding  a  certain  limit  (per  square  inch),  their  friction 
is  the  same  whatever  may  be  the  extent  of  their  surfaces  of 
contact. 

4thly.  That  the  friction  of  motion  is  wholly  independent 
of  the  velocity  of  the  motion.* 

Stilly.  That  where  unguents  are  interposed,  the  co-efficient 
of  friction  depends  upon  the  nature  of  the  unguent,  and  upon 
the  greater  or  less  abundance  of  the  supply.  In  respect  to 
the  supply  of  the  ungent,  there  are  two  extreme  cases,  that 
in  which  the  surfaces  of  contact  are  but  slightly  rubbed  with 
the  unctuous  matterf ,  and  that  in  which,  by  reason  of  the 
abudant  supply  of  the  unguent,  its  viscous  consistency,  and 
the  extent  of  the  surfaces  of  contact  in  relation  to  the  insist- 
ent pressure,  a  continuous  stratum  of  unguent  remains  con- 
tinually interposed  between  the  moving  surfaces,  and  the 
friction  is  thereby  diminished,  as  far  as  it  is  capable  of  being 
diminished,  by  the  interposition  of  the  particular  unguent 
used.  In  this  state  the  amount  of  friction  is  found  (as  might 
be  expected)  to  be  dependent  rather  upon  the  nature  of  the 
unguent  than  upon  that  of  the  surfaces  of  contact ;  accord- 
ingly M.  Morin,  from  the  comparison  of  a  great  number  of 
results,  has  arrived  at  the  following  remarkable  conclusion, 
easily  fixing  itself  in  the  memory,  and  of  great  practical 
value  : — "  that  with  unguents,  hog's  lard  and  olive  oil,  inter- 
posed in  a  continuous  stratum  between  them,  surfaces  of  wood 
on  metal,  wood  on  wood,  metal  on  wood,  and  metal  on  metal 
(when  in  motion),  have  all  of  them  very  nearly  the  same  co- 
efficient of  friction,  the  value  of  that  co-efficient  being  in  all 
cases  included  between  '07  and  '08. 

"  For  the  unguent  tallow,  the  co-efficient  is  the  same  as  for 
the  other  unguents  in  every  case,  except  in  that  of  metals  upon 
metals.  This  unguent  appears,  from  the  experiments  of  Mo- 
rin,  to  be  less  suited  to  metallic  substances  than  the  others, 
and  gives  for  the  mean  value  of  its  co-efficient  under  the  same 
circumstances  -10." 


134.  Whilst  there  is  a  remarkable  uniformity  in  the  results 
thus  obtained  in  respect  to  the  friction  of  surfaces,  between 
which  a  perfect  separation  is  effected  throughout  their  whole 
extent  by  the  interposition  of  a  continuous  stratum  of  the 

*  This  result,  of  so  much  importance  in  the  theory  of  machines,  is  fully  esta- 
blished by  the  experiments  of  Morin. 

\  As,  for  instance,  with  an  oiled  or  greasy  cloth. 


FRICTION.  127 

unguent,  there  is  an  infinite  variety  in  respect  to  those  states 
of  unctuosity  which  occur  between  the  extremes,  of  which 
we  have  spoken,  of  surfaces  merely  unctuous*  and  the  most 
perfect  state  of  lubrication  attainable  by  the  interposition 
of  a  given  unguent.  It  is  from  this  variety  of  states  of  the 
unctuosity  of  rubbing  surfaces,  that  so  great  a  discrepancy 
has  been  found  in  the  experiments  upon  friction  with  ungu- 
ents, a  discrepancy  which  has  not  probably  resulted  so  much 
from  a  difference  in  the  quantity  of  the  unguent  supplied  to 
the  rubbing  surfaces  in  different  experiments,  as  in  a  diffe- 
rence of  the  relation  of  the  insistent  pressures  to  the  extent 
of  rubbing  surface.  It  is  evident,  that  for  every  description 
of  unguent  there  must  correspond  a  certain  pressure  per 
square  inch,  under  which  pressure  a  perfect  separation  of 
two  surfaces  is  made  by  the  interposition  of  a  continuous 
stratum  of  that  unguent  between  them,  and  which  pressure 
per  square  inch  being  exceeded,  that  perfect  separation  can- 
not be  attained,  however  abundant  may  be  the  supply  of  the 
unguent. 

The  ingenious  experiments  of  Mr.  Nicholas  Woodf,  con- 
firmed by  those  of  Mr.  G.  Rennie^,.  have  fully  established 
these  important  conditions  of  the  friction  of  unctuous  surfaces. 
It  is  much  to  be  regretted  that  we  are  in  possession  of  no 
experiments  directed  specially  to  the  determination  of  that 
particular  pressure  per  square  inch,  which  corresponds  in 
respect  to  each  unguent  to  the  state  of  perfect  separation, 
and  to  the  determination  of  the  co-efficients  of  frictions  in 
those  different  states  of  separation  which  correspond  to  pres- 
sures higher  than  this. 

It  is  evident,  that  where  the  extent  of  the  surface  sustain- 
ing a  given  pressure  is  so  great  as  to  make  the  pressure  per 
square  inch  upon  that  surface  less  than  that  which  corres- 
ponds to  the  state  of  perfect  separation,  this  greater  extent  of 
surface  tends  to  increase  the  friction  by  reason  of  that  adhe- 
siveness of  the  unguent,  dependent  upon  its  greater  or  less 
viscosity,  whose  effect  is  proportional  to  the  extent  of  the 
surfaces  between  wThich  it  is  interposed.  The  experiments 
of  Mr.  Wood§  exhibit  the  effects  of  this  adhesiveness  in  a 
remarkable  point  of  view. 

*  Or  slightly  rubbed  with  the  unguent. 

t  Treatise  on  Rail-roads,  3rd  ed.  p.  399. 

i  Trans.  Royal  Soc.  1829. 

5  It  is  evident  that,  whilst  by  extending  the  unctuous  surface  which  sustains 
any  given  pressure,  we  diminish  the  co-efficient  of  friction  up  to  a  certain 
limit,  we  at  the  same  time  increase  that  adhesion  of  the  surfaces  which  results 


128  FRICTION. 

It  is  perhaps  deserving  of  enquiry,  whether  in  respect  to 
those  considerable  pressures  under  which  the  parts  of  the 
larger  machines  are  accustomed  to  move  upon  one  another, 
the  adhesion  of  the  unguent  to  the  surfaces  of  contact,  and 
the  opposition  presented  to  their  motion  by  its  viscidity,  are 
causes  whose  influence  may  be  altogether  neglected  as  com- 
pared with  the  ordinary  friction.  In  the  case  of  lighter 
machinery,  as  for  instance  that  of  clocks  and  watches,  these 
considerations  evidently  rise  into  importance. 


135.  The  experiments  of  M.  Morin  show  the  friction  of 
two  surfaces  which  have  been  for  a  considerable  time  in  con- 
tact,  to  be  not  only  different  in  its  amount  from  the  friction 
of  surfaces  in  continuous  motion,  but  also,  especially  in  this, 
that  the  laws  of  friction  (as  stated  above)  are,  in  respect  to 
the  friction  of  quiescence,  subject  to  causes  of  variation  and 
uncertainty  from  which  the  friction  of  motion  is  exempt. 
This  variation  does  not  appear  to  depend  upon  the  extent  of 
the  surfaces  of  contact,  in  which  case  it  might  be  referred  to 
adhesion  ;  for  with  different  pressures  the  co-efficient  of  the 
friction  of  quiescence  was  found,  in  certain  cases,  to  vary 
exceedingly,  although  the  surfaces  of  contact  remained  the 
same.*  The  uncertainty  which  would  have  been  introduced 
into  every  question  of  construction  by  this  consideration,  is 
removed  by  a  second  very  important  fact  developed  in  the 
course  of  the  same  experiments.  It  is  this,  that  by  the 
slightest  jar  or  shock  of  two  bodies  in  contact,  their  friction 
is  made  to  pass  from  that  state  which  accompanies  quiescence 

from  the  viscosity  of  the  unguent,  so  lhat  there  may  be  a  point  where  the  gain 
on  the  one  hand  begins  to  be  exceeded  by  the  loss  on  the  other,  and  where 
the  surface  of  minimum  resistance  under  the  given  pressure  is  therefore 
attained.  • 

Mr.  Wood  considers  the  pressure  per  square  inch,  which  corresponds  to  the 
minimum  resistance,  to  be  90lbs.  in  the  case  of  axles  of  wrought  iron  turning 
upon  cast  iron,  with  fine  neat's  foot  oil.  The  experiments  of  Mr.  Wood,  whilst 
they  place  the  general  results  stated  above  in  full  evidence,  can  scarcely  how- 
ever be  considered  satisfactory  as  to  the  particular  numerical  values  of  the  con- 
stants sought  in  this  inquiry.  In  those  experiments,  and  in  others  of  the  same 
class,  the  amount  of  friction  is  determined  from  the  observed  space  or  time 
through  which  a  body  projected  with  a  given  velocity  moves  before  all  its 
velocity  is  destroyed,  that  is,  before  its  accumulated  work  is  exhausted.  This 
is  an  easy  method  of  experiment,  but  liable  to  many  inaccuracies.  It  is  much 
to  be  regretted  that  the  experiments  of  Morin  were  not  extended  to  the  fric- 
tion of  unctuous  surfaces,  reference  being  had  to  the  pressure  per  square 
inch. 

*  Thus  in  the  case  of  oak  upon  oak  with  parallel  fibres,  the  co-efficient  of 
the  friction  of  quiescence  varied,  under  different  pressures  upon  the  same  sur- 
face, from  -55  to  '76. 


FRICTION.  129 

to  that  which  accompanies  motion ;  and  as  every  machine  or 
structure,  of  whatever  kind,  may  be  considered  to  be  subject 
to  such  shocks  or  imperceptible  motions  of  its  surfaces  of 
contact,  it  is  evident  that  the  state  of  friction  to  be  made 
the  basis  on  which  all  questions  of  statics  are  to  be  deter- 
mined, should  be  that  which  accompanies  continuous  motion. 
The  laws  stated  above  have  been  shown,  by  the  experiments 
of  Morin,  to  obtain,  in  respect  to  that  friction  which  accom- 
panies motion,  with  a  precision  and  uniformity  never  before 
assigned  to  them ;  they  have  given  to  all  our  calculations  in 
respect  to  the  theory  of  machines  (whose  moving  surfaces 
have  attained  their  proper  bearings  and;been  worn  to  their 
natural  polish)  a  new  and  unlooked-for  certainty,  and;  may 
probably  be  ranked  amongst  the  most  accurate  and:valuable- 
of  the  constants  of  practical  science. 

It  is,  however,  to  be  observed,  that  all  these  experiments; 
were  made  under  comparatively  small  insistent  pressures  a& 
compared  with  the  extent  of  the  surface  pressed  (pressures, 
not  exceeding  from  one  to  two  kilogrammes  per  square-  cen>- 
timeter,  or  from  about  14*3  to  28*6  Ibs.  per  square 'inch:.}  In 
adopting  the  results  of  M.  Morin,  it  is  of  importance  to  bear 
this  fact  in  mind,  because  the  experiments  of  Coulomb,  and 
particularly  the  excellent  experiments  of  Mr.  G\  Rennie,  car- 
ried far  beyond  these  limits  of  insistent  pressure*,  have  fully 
shown  the  co-efficient  of  the  friction  of  quiescence  to  increase 
rapidly,  from  some  limit  attained  long  before  the  surfaces 
abrade.  In  respect  to  some  surfaces,  as,  for  instance,  wrought 
iron  upon  wrought  iron,  the  co-efficient  nearly  tripled  itself 
as  the  pressure  advanced  to  the  limits  of  abrasion.  It  is 
greatly  to  be  regretted  that  no  experiments  have  yet  been 
directed  to  a  determination  of  the  precise  limit  about  which 
this  change  in  the  value  of  the  co-efficient  begins  to  take 
place.  It  appears,  indeed,  in  the  experiments  of  Mr.  Ren- 
nie  in  respect  to  some  of  the  soft  metals,  as,  for  instance,  tin 
upon  tin,  and  tin  upon  cast  iron ;  but  in  respect  to  the  harder 
metals,  his  experiments  passing  at  once  from  a  pressure  of 
32  Ibs.  per  square  inch  to  a  pressure  of  1*66  cwt.  per  square 
inch,  and  the  co-efficient  (in  the  case  of  wrought  iron  for  in- 
stance) from  about  -148  to  '25,  the  limit  which  we  seek  is 
lost  in  the  intervening  chasm.  The  experiments  of  Mr.  Ren- 
nie  have  reference,  nowever,  only  to  the  friction  of  qui- 
escence. It  seems  probable  that  the  co-efficient  of  the  fric- 

*  Mr.  Rennie's  experiments  were  carried,  in  some  cases,  to  from  5  cwt.  to 
7  cwt.  per  square  inch. 

9 


130  FRICTION. 

tion  of  motion  remains  constant  tinder  a  wider  range  of  pres- 
sure than  that  of  quiescence.  It  is  moreover  certain,  that 
the  limits  of  pressure  beyond  which  the  surfaces  of  contact 
begin  to  destroy  one  another  or  to  abrade,  are  sooner  reached 
when  one  of  them  is  in  motion  upon  the  other,  than  when 
they  are  at  rest:  it  is  also  certain  that  these  limits  are  not  in- 
dependent  of  the  velocity  of  the  moving  surface.  The  dis- 
cussion of  this  subject,  as  it  connects  itself  especially  with 
the  friction  of  motion,  is  of  great  importance  ;  and  it  is  to  be 
regretted,  that,  with  the  means  so  munificently  placed  at  his 
disposal  by  the  French  Government,  M.  Morin  did  not  ex- 
tend his  experiments  to  higher  pressures,  and  direct  them 
more  particularly  to  the  circumstances  of  pressure  and  velo- 
city under  which  a  destruction  of  the  rubbing  surfaces  first 
begins  to  show  itself,  and  to  the  amount  of  the  destruction 
of  surface  or  wear  of  the  material  which  corresponds  to  the 
same  space  traversed  under  different  pressures  and  different 
velocities.  Any  accurate  observer  who  should  direct  his 
attention  to  these  subjects  would  greatly  promote  the  inter- 
ests of  practical  science. 


SUMMARY  OF  THE  LAWS  OF  FRICTION. 

136.  From  what  has  here  been  stated  it  results,  that  if  P 
represent  the  perpendicular  or  normal  force  by  which  one 
body  is  pressed  upon  the  surface  of  another,  F  the  friction  of 
the  two  surfaces,  or  the  force,  which  being  applied  parallel 
to  their  common  surface  of  contact,  would  cause  one  of  them 
to  slip  upon  the  surface  of  the  other,  and/*  the  co-efficient  of 
friction,  then,  in  the  case  in  which  no  unguent  is  interposed, 
f  represents  a  constant  quantity,  and  (Art.  133.) 

F=/P (109); 

a  relation  which  obtains  accurately  in  respect  to  the  friction 
of  motion,  and  approximately  in  respect  to  the  friction  of 
quiescence. 

137.  The  same  relation  obtains,  moreover,  in  respect  to 
unctuous  surfaces  when  merely  rubbed  with  the  unguent,  or 
where  the  presence  of  the  unguent  has  no  other  influence 
than  to  increase  the  smoothness  of  the  surfaces  of  contact 
without  at  all  separating  them  from  one  another. 

In  unctuous  surf aces  partially  lubricated,  or  between  which 


THE   LIMITING    ANGLE   OF   RESISTANCE.  131 

a  stratum  of  unguent  is  partially  interposed,  the  co-efficient 
of  friction/1  is  dependent  for  its  amount  upon  the  relation  of 
the  insistent  pressure  to  the  extent  of  the  surface  pressed, 
or  upon  the  pressure  per  square  inch  of  surface.  This 
amount,  corresponding  to  each  pressure  per  square  inch  in 
respect  to  the  different  unguents  used  in  machines,  has  not 
yet  been  made  the  subject  of  satisfactory  experiments. 

The  amount  of  the  resistance  F  opposed  to  the  sliding  of 
the  surfaces  upon  one  another  is,  moreover,  as  well  in  this 
case  as  in  that  of  surfaces  perfectly  lubricated,  influenced  by 
the  adhesiveness  of  the  unguent,  and  is  therefore  dependent 
upon  the  extent  of  the  adhering  surface ;  so  that,  if  S  repre- 
sent the  number  of  square  units  in  this  surface,  and  a  the 
adherence  of  each  square  unit,  then  aS  represents  the  whole 
adherence  opposed  to  the  sliding  of  the  surfaces,  and 


(110); 

where  f  is  a  function  of  the  pressure  per  square  unit  ^-,  and 

a  is  an  exceedingly  small  factor  dependent  on  the  viscosity 
of  the  unguent. 

THE  LIMITING  ANGLE  OF  RESISTANCE. 

We  shall,  for  the  present,  suppose  the  parts  of  a  solid  body 
to  cohere  so  firmly,  as  to  be  incapable  of  separation  by  the 
action  of  any  force  which  may  be  impressed  upon  them. 
The  limits  within  which  this  suposition  is  true  will  be  dis- 
cusse,d  hereafter. 

It  is  not  to  this  resistance  that  our  present  inquiry  has 
reference,  but  to  that  which  results  from  the  friction  of  the 
surface  of  bodies  on  one  another,  and  especially  to  the  direc- 
tion of  that  resistance. 


138.  Any  pressure  applied  to  the  surface  of  an  immoveable 
solid  body  by  the  intervention  of  another  body  moveable 
upon  it,  will  be  sustained  by^  the  resistance  of  t/ie  surfaces 
of  contact,  whatever  be  its  direction,  provided  only  the  an- 
gle which  that  direction  makes  with  the  perpendicular  to 
the  surfaces  of  contact  do  not  exceed  a  certain  angle  called 

the     LIMITING     ANGLE     OF     RESISTANCE     of   those    SURFACES. 


132  THE   LIMITING    ANGLE   OF   RESISTANCE. 

This  is  true,  however  great  the  pressure  may  he.  Also,  if 
the  inclination  of  the  pressure  to  the  perpendicular  exceed 
the  limiting  angle  of  resistance,  then  this  pressure  will  not 
he  sustained  by  the  resistance  of  the  surfaces  of  contact  y 
and  this  is  true,  however  small  the  pressure  may  ~be. 

Let  PQ  represent  the  direction  in  which  the  surfaces  of 
two  bodies  are  pressed  together  at  Q,  and  let 
QA  be  a  perpendicular  or  normal  to  the  sur- 
faces of  contact  at  that  point,  then  will  the  pres- 
sure PQ  be  sustained  by  the  resistance  of  the 
surfaces,  however  great  it  may  be,  provided  its 
direction  lie  within  a  certain  given  angle  AQB, 
called  the  limiting  angle  of  resistance  ;  and  it  will  not  be  sus- 
tained, however  small  it  may  be,  provided  its  direction  lie 
without  that  angle.  For  let  this  pressure  be  represented  by 
PQ,  and  let  it  be  resolved  into  two  others  AQ  and  RQ,  of 
which  AQ  is  that  by  which  it  presses  the  surfaces  together 
perpendicularly,  and  RQ  that  by  which  it  tends  to  cause 
them  to  slide  upon  one  another,  if  therefore  the  friction  F 
produced  by  the  first  of  these  pressures  exceed  the  second 
pressure  RQ,  then  the  one  body  will  not  be  made  to  slip 
upon  the  other  by  this  pressure  PQ,  however  great  it  may 
be ;  but  if  the  friction  F,  produced  by  the  perpendicular 
pressure  AQ,  be  less  than  the  pressure  RQ,  then  the  one 
body  will  be  made  to  slip  upon  the  other,  however  small  PQ 
may  be.  Let  the  pressure  in  the  direction  PQ  be  repre- 
sented by  P,  and  the  angle  AQP  by  6,  the  perpendicular 
pressure  in  AQ  is  then  represented  by  P  cos.  d,  and  therefore 
the  friction  of  the  surfaces  of  contact  by/T  cos.  0,  f  repre- 
senting the  co-efficient  of  friction  (Art.  136.).  Moreover,  the 
resolved  pressure  in  the  direction  RQ  is  represented  by  P 
sin.  &.  The  pressure  P  will  therefore  be  sustained  by  the 
friction  of  the  surfaces  of  contact  or  not,  according  as 

P  sin.  &  is  less  or  greater  than  fP  cos.  6 ; 

or,  dividing  both  sides  of  this  inequality  by  P  cos.  d,  ac 
cording  as 

tan.  6  is  less  or  greater  than  f. 

Let,  now,  the  angle  AQB  equal  that  angle  whose  tangent  is 
f,  and  let  it  be  represented  by  0,  so  that  tan.  0=/".     Substi- 
tuting this  value  of  f  in  the  last  inequality,  it  appears  that 
the  pressure  P  will  be  sustained  by  the  friction  of  the  s^ 
faces  of  contact  or  not,  according  as 


THE   TWO    STATES   BORDERING   UPON   MOTION.  133 

tan.  &  is  greater  or  less  than  tan.  0, 
that  is,  according  as 

6  is  less  or  greater  than  0, 
or  according  as 

AQP  is  less  or  greater  than  AQB. 
Therefore,  &c.  [Q.  E.  D.] 

THE  CONE  OF  RESISTANCE. 

139.  If  the  angle  AQB  be  conceived  to  revolve  about  the 
axis  AQ,  so  that  BQ  may  generate  the  surface  of 
a  cone  BQC,  then  this  cone  is  called  the  CONE  OP 
RESISTANCE  i  it  is  evident,  that  any  pressure,  how- 
ever great,  applied  to  the  surfaces  of  contact  at 
Q  will  be  sustained  by  the  resistance  of  the  sur- 
faces of  contact,  provided   its  direction  be  any 
where  within  the  surface  of  this  cone ;  and  that  it  will  not 
be  sustained,  however  small  it  may  be,  if  its  direction  lie  any 
where  without  it. 


THE  Two  STATES  BORDERING  UPON  MOTION. 

140.  If  the  direction  of  the  pressure  coincide  with  the  sur- 
face of  the  cone,  it  will  be  sustained  by  the  friction  of  the 
surfaces  of  contact,  but  the  body  to  which  it  is  applied  will 
be  upon  the  point  of  slipping  upon  the  other.  The  state  of 
the  equilibrium  of  this  body  is  then  said  to  be  that  BORDER- 
ING UPON  MOTION.  If  the  pressure  P  admit  of  being  applied 
in  any  direction  about  the  point  Q,  there  are  evidently  an 
infinity  of  such  states  of  the  equilibrium  bordering  upon  mo- 
tion, corresponding  to  all  the  possible  positions  of  P  on  the 
surface  of  the  cone. 

If  the  pressure  P  admit  of  being  applied  only  in  the  same 
plane,  there  are  but  two  such  states,  corresponding  to  those 
directions  of  P,  which  coincide  with  the  two  intersections  of 
this  plane  with  the  surface  of  the  cone  ;  these  are  called  the 
superior  and  inferior  states  bordering  upon  motion.  In  the 
case  in  which  the  direction  of  P  is  limited  to  the  plane  AQB, 
BQ  and  CQ  represent  its  directions  corresponding  to  the 


134  THE   TWO    STATES    BORDERING   UPON    MOTION. 

two  states  bordering  on  motion.  Any  direction  of  P  within 
the  angle  BQC  corresponds  to  a  state  of  equilibrium ;  any 
direction,  without  this  angle,  to  a  state  of  motion. 


141.  Since,  when  the  direction  of  the  pressure  P  coincides 
with  the  surface  of  the  cone  of  resistance,  the  equilibrium  is 
in  the  state  bordering  upon  motion  ;  it  follows,  conversely, 
and  for  the  same  reasons,  that  this  is  the  direction  of  the 
pressure  sustained  by  the  surfaces  of  contact  of  two  bodies 
whenever  the  state  of  their  equilibrium  is  that  bordering  upon 
motion.  This  being,  moreover,  the  direction  of  the  pressure 
of  the  one  body  upon  the  other  is  manifestly  the  direction  of 
the  resistance  opposed  by  the  second  body  to  the  pressure  of 
the  first  at  their  surface  of  contact,  for  this  single  pressure 
and  this  single  resistance  are  forces  in  equilibrium,  and  there- 
fore equal  and  opposite.  All  that  has  been  said  above  of  the 
single  pressure  and  the  single  resistance  sustained  by  two 
surfaces  of  contact,  is  manifestly  true  of  the  resultant  of  any 
number  of  such  pressures,  and  of  the  resultant  of  any  num- 
ber of  such  resistances.  Thus  then  it  follows,  that  when  any 
number  of  pressures  applied  to  a  body  movedble  upon  another 
which  is  fixed,  are  sustained  by  the  resistance  of  the  surfaces 
of  contact  of  the  two  bodies,  and  are  in  the  state  of  equilibrium 
bordering  upon  motion,  then  the  direction  of  the  resultant  of 
these  pressures  coincides  with  the  surface  of  the  cone  of  resist- 
ance, as  does  that  also  of  the  resultant  of  the  resistances  of  the 
different  points  of  the  surfaces  of  contact*,  that  is,  they  are 
both  inclined  to  the  perpendicular  to  the  surfaces  of  contact 
(at  the  point  where  they  intersect  it),  at  an  angle  equal  to  the 
limiting  angle  of  resistance. 


*  The  properties  of  the  limiting  angle  of  resistance  and  the 
ance,  were  first  given  by  the  author  of  this  work  in  a  paper  published  m  the 
Cambridge  Philosophical  Transactions,  vol.  v. 


FRICTION. 


135 


TABLE  I. 
Friction  of  Plane  Surfaces,  when  they  have 


some  time  in  Contact. 


Surfaces  in  Contact. 

Disposition  of 
the  Fibres. 

State  of  the 
Surfaces. 

Co-efficient 
of  Friction. 

Limiting 
Angle  of 
Resist- 
ance. 

EXPERIMENTS  OF  M.  MORIN. 

parallel 

without 
unguent 

0-62 

31°  48' 

ditto 

rubbed  with 
dry  soap 

0-44 

23     45 

Oak  noon  oak 

perpendicu- 
lar 
ditto 

without         ) 
unguent    f 
with  water 

0-54  ' 
0-71 

28     22 
35     23 

endways  of  "] 

one  upon  | 
the      flat  V 

without         ) 

0'43 

23     16 

surface  of 

unguent    ) 

the  other  J 

Oak  upon  elm 

parallel 

ditto 

0-38 

20     49 

C 

ditto 

ditto 

0-69 

34     37 

Elm  upon  oak         -         < 

ditto 

rub  tfed  with 
dry  soap 

0'41 

22     18 

V 

perpendicu- 

without 

0'57 

29     41 

\ 

lar 

unguent 

Ash,  fir,  beech,  service-  ) 
tree,  upon  oak               \ 

parallel 

ditto 

0-53 

27     56 

\ 

the  leather 
flat 

ditto 

0-61 

31     23 

Tanned  leather  upon  oak^ 

the   leather  1 
length-      1 

ditto 

0-43 

23     16 

'          1 

ways,  but  f 
sideways  J 

steeped    in  ) 
water        j 

0-79 

38     19 

„,     ,         fupon  a  plane 

B*ack    .         surface     of  - 
dressed        Qak 

parallel         j 

without         ) 
unguent    ) 

0-74 

36     30 

leatheM  upon  a  round- 
er strap       ^d    gurface 

leather            f      , 

perpendicu-  ) 
lar              J 

ditto 

0-47 

25     11 

1      oi  oak 

Hemp  matting  upon  oak  -I 

parallel 
ditto 

ditto 
steeped    in  ) 
water        J 

0-50 

0-87 

26     34 
41       2 

Hemp  cords  upon  oak    - 

ditto 

without         ) 
unguent    ) 

0-80 

38     40 

ditto 

0-62 

31     48 

Iron  upon  oak 

ditto 

steeped    in 
water 

0-65 

33       2 

Cast-iron  upon  oak 

ditto 

ditto 

0-65 

33       2 

Copper  upon  oak    - 

ditto             | 

without 
unguent 

0-62 

31     48 

| 

steeped    in 

0-62 

31     48 

Ox-hide  as  a  piston  sheath  ) 

flat  or  side-] 

water 
with        oil, 

upon  cast-iron               ) 

ways 

tallow,  or 

0'12 

6     51 

hog's  lard 

136 


FRICTION. 


Limiting 

Surfaces  in  Contact. 

Disposition  of 
the  Fibres. 

State  of  the 
Surfaces. 

Co-efficien 
of  Friction 

Angle  of 
Resist- 

ance. 

['EXPERIMENTS  OF  M.  MORIN. 

—  continued. 

Black  dressed  leather,  or  \ 
strap  leather,  upon  a  > 
cast-iron  pulley             ) 

flat                ) 

without         ) 
unguent    ) 
steeped 

0-28 
0-38 

15°  39' 
20    49 

Cast-iron  upon  cast-iron  - 

ditto             -j 

without         ) 
unguent    j" 

0-16 

9       6 

Iron  upon  cast-iron 

ditto 

ditto 

0-19 

10    46 

Oak,  elm,  yoke  elm,  iron,  "| 
cast-iron,    and    brass  1 
sliding  two  and  two,  [ 
one  upon  another         J 

ditto 

with  tallow 
with  oil,  or  ) 
hog's  lard  j" 

o-iof 

0-15^: 

5     43 
8     32 

Calcareous  oolite    stone  ) 
upon  calcareous  oolite  ) 

ditto 

without         ) 
unguent    j 

0-74 

36     30 

Hard   calcareous   stone,  ) 

called      muschelkalk,  > 

ditto 

ditto 

0-75 

36     52 

upon  calcareous  oolite  ) 

Brick    upon    calcareous  ) 
oolite                              J 

ditto 

ditto 

0-67 

33     50 

Oak  upon  calcareous     j 
oolite                             ( 

wood    end-  ) 
ways         f 

ditto 

0-63 

32     13 

•Iron  upon  calcareous  oolite 

flat 

ditto 

0-49 

26       7 

i  Hard  calcareous  stone,  or 

muschelkalk,        upon  • 

ditto 

ditto 

0'70 

35       0 

muschelkalk 

'Calcareous   oolite  stone 
upon  muschelkalk 

ditto 

ditto 

0-75 

36     52 

Brick  upon  muschelkalk  - 

ditto 

ditto 

0-67 

33     50 

Iron  upon  muschelkalk    • 

ditto 

ditto 

0-42 

22     47 

Oak  upon  muschelkalk    - 

ditto 

ditto 

0-64 

32     38 

with  a  coat-  "1 

ing  of  mor- 

Calcareous  oolite  stone  ) 
upon  calcareous  oolite  J" 

ditto 

tar,ofthree 
parts  of  fine  V 
sand      and 

0'74§ 

36     80 

one  part  of 

slack  lime   J 

*  The  surfaces  retaining  some  unctuousness. 

f  When  the  contact  has  not  lasted  long  enough  to  express  the  grease. 
\  When  the  contact  has  lasted  long  enough  to  express  the  grease  and  bring 
back  the  surfaces  to  an  unctuous  state. 

§  After  a  contact  of  from  ten  to  fifteen  minutes. 


FRICTION. 


137 


Nature  of  Bodies  and  Unguents. 

Co-efficient 
of  Friction. 

Limiting 
Angle. 

Soft  calcareous  stone,  well  dressed,  upon  the  same 

0-74 

36°  30' 

Hard  calcareous  stone,  ditto        .... 

0-75 

36     52 

Common  brick,  ditto        - 

0-67 

33     50 

Oak,  endways,  ditto         ..... 

0-63 

32     13 

Wrought  iron,  ditto         ..... 

0-49 

26       7 

Hard  calcareous  stone,  well  dressed,  upon  hard  calcare- 

ous stone           ...... 

0-70 

35       0 

Soft,  ditto             ...... 

0-75 

36     52 

Common  brick,  ditto        ..... 

0-67 

83     50 

Oak,  endways,  ditto          • 

0-64 

32     37 

Wrought  iron,  ditto         ..... 

0-42 

22    47 

Soft  calcareous  stone  upon  soft  calcareous  stone,  with 

fresh  mortar  of  fine  sand         .... 

0-74 

36     30 

EXPERIMENTS  BY  DIFFERENT  OBSERVERS. 

Smooth  free-stone  upon  smooth  free-stone,  dry  (Rennie) 

0-71 

35     23 

Ditto,  with  fresh  mortar  (Rennie) 

0-66 

33     26 

Hard  polished  calcareous  stone  upon  hard  polished  cal- 

careous stone    -                          .... 

0-58 

30      7 

Calcareous  stone  upon  ditto,  both  surfaces  being  made 

rough  with  a  chisel  (Bonchardi)           ... 

0-78 

37     58 

Well  dressed  granite  upon  rough  granite  (Rennie) 

0-66 

33     26 

Ditto,  with  fresh  mortar,  ditto  (Rennie)  - 

0-49 

26      7 

Box  of  wood  upon  pavement  (Regnier)  - 

0-58 

30      7 

Ditto  upon  beaten  earth  (Herbert)           - 

0-33 

18     16 

Libage  stone  upon  a  bed  of  dry  clay      - 

0-51 

27       2 

Ditto,  the  clay  being  damp  and  soft        ... 

0-34 

18    47 

Ditto,  the  clay  being  equally  damp,  but  covered  with 

thick  sand  (Greve)       - 

0-40 

21     48 

138 


FfilOTIOX. 


TABLE  II. 
friction  of  Plane  Surfaces,  in  Motion  one  upon  the  other. 


Surfaces  in  Contact. 

Disposition  of 
the  Fibres. 

State  of  the 
Surfaces. 

Co-efficient 
of  Friction. 

Limiting 
Angle  of 
Resist- 
ance. 

EXPERIMENTS  OP  M.  MORIN. 

§ 

parallel 

without 
unguent 

0-48 

25C39' 

> 

ditto 

rubbed  with 

j 

0-16 

9     6 

dry  soap 

perpendicu- 
lar 

without 
unguent 

0-34 

18  47 

Oak  upon  oak 

ditto 

steeped    in 

0-25 

14    3 

water 

wood    endO 

ways    on 
wood         j- 
length- 

without         ) 
unguent    J 

0-19 

10  46 

ways        J 

r 

parallel 

ditto 

0-43 

23  17 

Ehn  upon  oak         -        •< 

perpendicu-  ) 
lar             J 

ditto 

0-45 

24  14 

[ 

parallel 

ditto 

0-25 

14    3 

Ash,  fir,  beech,  wild  pear-  } 
tree,  and  service-tree,  V 
upon  oak                        ) 

ditto 

ditto 

0-36  to 
0-40 

)  19  48 
}21  49 

ditto 

0-62 

31  48 

with  water 

0-26 

14  35 

Iron  upon  oak 

ditto 

rubbed  with 
dry   soap 

0-21 

11  52 

without 

unguent 

0'49 

26     7 

with  water 

0-22 

12  25 

Cast-iron  upon  oak 

ditto 

rubbed  with 
dry  soap 

0-19 

10  46 

Copper  upon  oak   - 

ditto             j 

without 
'  unguent 

0-62 

31  48 

Iron  upon  elm 

ditto 

ditto 

0-25 

14     3 

Cast-iron  upon  elm  - 

ditto 

ditto 

0-20 

11  19 

Black    dressed    leather  ) 
upon  oak 

ditto 

ditto 

0-27 

15    7 

r 

flat,  or         1 

Tanned  leather  upon  oak  j 

**u>     v* 

length-      1 
ways,  and  ( 
edgeways  J 

ditto             -j 
with  water 

3-30  to 
0-35 
0-29 

16  42 
19  18 
16  11 

' 

without 

unguent 

0-56 

29  15 

Tanned     leather     upon  ) 
cast-iron  and  brass        J 

ditto 

steeped    in 
water 
greased  and 

0-36 

19  48 

steeped  in  • 

0-23 

12  58 

water 

. 

. 

with  oil 

0-15 

8  32 

FRICTION. 


139 


Surfaces  in  Contact. 

Disposition  of 
the  Fibres. 

State  of  the 
Surfaces. 

Coefficient 
of  Friction. 

Limiting    i 
Angle  of   i 
Resist- 
ance. 

EXPERIMENTS  OF  M.  MORIN. 

—  continued. 

Hemp,  in  threads  or  in] 

parallel 

without 
unguent 

0-52 

27°29' 

cord,  upon  oak 

perpendicu- 
lar 

i 

with  water 

0-33 

18  16 

Oak  and  elm  upon  cast-  \ 
iron                                 ) 

parallel 

without 
unguent 

0-38 

20  49 

Wild  pear-tree,  ditto 

ditto 

ditto 

0-44 

23  45 

Iron  upon  iron 

ditto 

ditto 

0-44 

23  45* 

Iron  upon  cast-iron  and  ) 
brass                                 [ 

ditto 

ditto 

o-isf 

10  13 

Cast-iron,  ditto 

ditto 

ditto 

0-15 

8  32 

{upon  brass  - 

ditto 

ditto 

0-20 

11  19 

upon  cast-iron 

ditto 

ditto 

0-22 

12  25 

upon  iron    - 

ditto 

ditto 

0'16| 

9     6 

greased    in  ~ 

the  usual 

Oak,  elm,  yoke  elm,  wild  "1 
pear,  cast-iron,  wrought 
iron,  steel,  and  moving  V 

ditto 

way  with 
tallow, 
hog's  lard, 
oil      soft 

0-07  to 
0-08§ 

U       1 
14     35 

one  upon  another,  or  on 
themselves 

gom 
slightly         ; 

greasy  to 

w 

0-15 

8  32 

the  touch 

Calcareous  oolite    stone 
upon  calcareous  oolite 

^ 

without 
unguent 

: 

0-64 

82  37 

Calcareous  stone,  called 

muschelkalk,  upon  cal-  • 

ditto 

ditto 

0-67 

33  50 

careous  oolite 

Common  brick  upon  cal- 

ditto 

ditto 

0-65 

33     2 

careous  oolite 

Oak   upon   calcareous      j 
oolite                              \ 

wood    end- 
ways 

ditto 

0-38 

20  49 

Wrought  iron,  ditto 

parallel 

ditto 

0-69 

34  37 

Calcareous  stone,  called 

muschelkalk,upon  mus- 

ditto 

ditto 

0-38 

20  49 

chelkalk 

Calcareous   oolite   stone 
upon  muschelkalk 

ditto 

ditto 

0-65 

33     2 

Common  brick,  ditto 

ditto 

ditto 

0'60 

30  58 

Oak  upon  muschelkalk     \ 

wood    end- 
ways 

\ 

ditto 

0-38 

20  49 

i    ditto 

0-24 

Iron  upon  muschelkalk    - 

saturated 
with  water 

0-30 

16  42 

*  The  surfaces  wear  when  there  is  no  grease. 

•jr  The  surfaces  still  retaining  a  little  unctuousness.  \  Ibid. 

§  When  the  grease  is  constantly  renewed  and  uniformly  distributed,  thia 
proportion  can  be  reduced  to  0'05. 


140 


FEICTIOK. 


TABLE  III. 

friction  of  Gudgeons  or  Axle-ends,  in  Motion,  upon  their  Bearings. 
(From  the  experiments  of  Morin.) 


Surfaces  in  Contact. 

State  of  the  Surfaces. 

Co-efficient  of  Friction  when 
the  Grease  is  renewed. 

Limiting 
Angle  of 
Resistance. 

In  the  usual 
Way. 

Continuously. 

coated  with  oil  of 

(    4°  0' 

olives,  with  hog's 
lard,  tallow,  anc 

0-07  to  0-08 

0-054 

\    4  35 
(    3     6 

Cast-iron    axles 
in       cast-iron  - 
bearings 

soft  goin 
with     the     same, 
and  water 
coated    with     as- 

0-08 

0-28 

4  35 

phaltum 

0-054 

0-19 

3     6 

greasy 

0-14 

. 

7  58 

greasy  and  wetted 

0-14 

•              • 

7  58 

coated  with  oil  of 

(    4     0 

Cast-iron   axles, 
ditto 

olives,  with  hog's 
lard,  tallow,  and 
soft  gom 

0-07  to  0-08 

0-054 

\    4  35 
(    3     6 

greasy 

0-16 

- 

9     6 

greasy  and  damped 

0-16 

* 

9     6 

scarcely  greasy 

0-19 

10  46 

without  unguent 

0-18 

. 

10  12 

Cast-iron     axles 
in  lignum  vit,ae< 
bearings 

with  oil   or   hog's 
lard 
greasy  with  ditto 
greasy,      with      a 
mixture  of  hog's 
lard  and  molyb- 

o-io 

I       0-14 

0-090 

5     9 
5  43 

7  58 

daena 

J 

Wrought-iron 
axles  in  cast-K 
iron  bearings 

coated     with      oil 
of  olives,  tallow, 
hog's     lard,     or 
soft  gom 

L  0-07  to  0-08 

0-054 

(    4     0 
\    4  35 
(    3     6 

' 

coated  with  oil  of 

) 

i4     0 
*         v 

olives,  hog's  lard, 

[•  0-07  to  0-08 

0-054 

4  35 

[ron     axles    in 
brass  bearings  " 

or  tallow 
coated   with   hard 
gom 

) 

[•        0-09 

. 

3     6 
5     9 

greasy  and  wetted 

0-19 

. 

10  46 

scarcely  greasy  • 

0-25 

* 

14     2 

[ron    axles      in 

coated    with     oil, 

)          nil 

lignum  vitae 

or  hog's  lard 

h       O'll 
\ 

*              - 

6  17 

bearings 

greasy 

0-19 

. 

10  46 

Brass    axles    in 

coated  with  oil 

o-io 

„ 

5  43 

brass  bearings 

with  hog's  lard 

0-09 

m 

5     9 

Brass    axles    in 
cast-iron  bear-  • 
ings 

coated     with     oil 
or  tallow 

0-045  to  0-052 

j    2  35 
{    2  59 

*  The  surfaces  beginning  to  wear. 


FRICTION. 


Co-efficient  of  Friction  when 

Surfaces  in  Contact. 

State  of  the  Surfaces. 

the  Grease  is  renewed. 

Limiting 
Angle  of 

In  the  usual 
Way. 

Continuously. 

Resistance. 

Lignum        vitse  j 
axles,  ditto      1 

coated  with  hog's 
lard 
greasy 

I        0-12 
0-15 

•                •» 

6051' 
8  82 

Lignum        vitas  "I 

axles    in    lig-  I 
num          vitas  j 

coated  with  hog's 
lard 

1-       - 

0-07 

4    0 

bearings          J 

TABLE  IV. 

Co-efficients  of  friction  under  Pressures  increased  continually  up  to  the 

Limits  of  Abrasion. 
(From  the  experiments  of  Mr.  G.  Rennie.*) 


Co-efficients  of  Friction. 

Pressure  per 
Square  Inch. 

Wrought-iron 
upon 
Wrought-iron. 

Wrought-iron 
upon 
Cast-iron. 

Steel  upon 
Cast-iron. 

Brass  upon 
Cast-iron. 

32-  51bs. 

•140 

•174 

•166 

•157 

1-66  cwt. 

•250 

•275 

•300 

•225 

2'00 

•271 

•292 

•333 

•219 

2-33 

•285 

•321 

•340 

•214 

2-66 

•297 

•329 

•344 

•211 

3-00 

•312 

•333 

•347 

•215 

3-33 

•350 

•351 

•351 

•206 

3-66 

•376 

•353 

•353 

•205 

4-00 

•376 

•365 

•354 

•208 

4-33 

•395 

•366 

•356 

•221 

4-66 

•403 

•366 

•357 

•223 

5-00 

•409 

•367 

•358 

•233 

5-33 

•367 

•359 

•234 

5-66 

•367 

•367 

•235 

6-00 

•376 

•408 

•283 

6-33 

•484 

•234 

6  -60 

•235 

7-00 

•282 

7-88 

•273 

Phil.  Trans.  1829,  table  8.  p.  159. 


142  THE   RIGIDITY   OF    COEDS. 


THE  RIGIDITY  OF  COEDS. 

• 

142.  It  is  evident  that,  by  reason  of  that  resistance  to 
r~ —  ,  deflexion  which  constitutes  the  ri- 
gidity of  a  cord,  a  certain  force  or 
pressure  must  be  called  into  action 
whenever  it  is  made  to  change  its 
rectilineal  direction,  so  as  to  adapt 
itself  to  the  form  of  any  curved  sur- 
face over  which  it  is  made  to  pass ; 
as,  for  instance,  over  the  circumfe- 
rence of  a  pulley  or  wheel.  Sup- 
pose such  a  cord  to  sustain  tensions  represented  by  Pa  and 
P2,  of  which  Px  is  on  the  point  of  preponderating,  and  let 
the  friction  of  the  axis  of  the  pulley  be,  for  the  present, 
neglected.  It  is  manifest  that,  in  order  to  supply  the  force 
necessary  to  overcome  the  rigidity  of  the  cord  and  to  pro- 
duce its  deflection  at  B,  the  tension  P1  must  exceed  P2 ; 
whereas,  if  there  were  no  rigidity,  P,  would  equal  P2 ;  so 
that  the  effect  of  the  rigidity  in  increasing  the  tension  P,  is 
the  same  as  though  it  had,  by  a  certain  quantity,  increased 
the  tension  P2.  Now,  from  a  very  numerous  series  of 
experiments  made  by  Coulomb  upon  this  subject,  it  appears 
that  the  quantity  by  which  the  tension  P2  may  thus  be  con- 
sidered to  be  increased  by  the  rigidity,  is  partly  constant 
and  partly  dependent  on' the  amount  of  P2;  so  as  to  be 
represented  by  an  algebraical  formula  of  two  terms,  one 
of  which  is  -i  constant  quantity,  and  the  other  the  product 
of  a  constant  quantity  by  P2.  Thus  if  D  represent  the 
constant  part  of  this  formula,  and  E  the  constant  factor 
of  P2,  then  is  the  effect  of  the  rigidity  of  the  cord  the  same 
as  though  the  tension  P2  were  increased  by  the  quantity 
D+E.P,. 

When  the  cord,  instead  of  being  bent,  under  different 
pressures,  upon  circular  arcs  of  equal  radii,  was  bent  upon 
circular  arcs  of  different  radii,  then  this  quantity  D  +  E .  P2 ; 
by  which  the  tension  P2  may  be  considered  to  be  increased 
by  the  rigidity,  was  found  to  vary  inversely  as  the  radii 
of  the  arcs ;  so  that,  on  the  whole,  it  may  be  represented 
by  the  formula 


THE   EIGIDITY   OF   COKDS.  143 

D+E  .  P, 


R 


(HI), 


where  E  represents  the  radius  of  the  circular  arc  over  which 
the  rope  is  bent.  Tims  it  appears  that  the  yielding  tension 
P2  may  be  considered  to  have  been  increased  by  the  rigidity 
of  the  rope,  when  in  the  state  bordering  upon  motion,  so  as 
to  become 


This  formula  applies  only  to  the  bending  of  the  same  cord 
under  different  tensions  upon  different  circular  arcs  :  for  dif- 
ferent cords,  the  constants  D  and  E  vary  (within  certain 
limits  to  be  specified)  as  the  squares  of  the  diameters  or  of  the 
circumferences  of  the  cords,  in  respect  to  new  cords,  wet  or 
dry  /  111  respect  to  old  cords  they  vary  nearly  as  the  power  f 
of  the  diameters  or  circumferences. 

Tables  have  been  furnished  by  Coulomb  of  the  values  of 
the  constants  I)  and  E.  These  tables,  reduced  to  English 
measures,  are  given  on  the  next  page.* 

*  The  rigidity  of  the  cord  exerts  its  influence  to  increase  resistance^  only  at 
that  point  where  the  cord  winds  upon  the  pulley  ;  at  the  point  where  it  leaves 
the  pulley  its  elasticity  favours  rather,  and  does  not  perceptibly  affect,  the 
conditions  of  the  equilibrium. 

In  all  calculations  of  machines,  in  which  the  moving  power  is  applied  by  the 
intervention  of  a  rope  passing  over  a  pulley,  one-half  the  diameter  of  rope  is 
to  be  added  to  the  radius  of  the  pulley,  or  to  the  perpendicular  on  the  direction 
of  the  rope  from  the  point  whence  the  moments  are  measured,  the  pressure 
applied  to  the  rope  producing  the  same  effect  as  though  it  were  all  exerted 
along  the  axis  of  the  rope.  > 


144 


THE   KIGIDITY   OF   CORDS. 


TABLE  V.    RIGIDITY  OF  ROPES. 

Table  of  the  values  of  the  constants  D  and  E,  according  to  the  experiments  of 
Coulomb  (reduced  to  English  measures}.  The  radius  R  of  the  pulley  is  to  be 
taken  in  feet. 

No.  1.     New  dry  cords.     Rigidity  proportional  to  the  square  of  the 
circumference. 


Circumference  of 
the  Rope  in  Inches. 

Value  of  D  in  Ibs. 

Value  of  E  in  Ibs. 

1 
2 
4 
8 

•131528 
•526108 
2-104451 
8-413702 

•033533 

•023030 
•073175 
•368494 

Squares  of  propor- 
tions of  the  in- 
termediate cir- 
cumferences to 
those  of  the 
table. 


No.  2.    New  ropes  dipped  in  water.     Rigidity  proportional 
to  the  square  of  the  circumference. 


Circumference  of 
the  Rope  in  Inches. 

Value  of  D  in  Ibs. 

Value  of  E  in  Ibs. 

1 
2 
4 

8 

•263053 
1-052217 

4-208902 
16-835606 

•0057576 
•0230303 
•0731755 
•3684860 

No.  3. 


Dry  half-worn  ropes.      Rigidity  proportional  to  the  square  root 
of  the  cube  of  the  circumference. 


Circumference  of 
the  Rope  in  Inches. 

Value  of  D  in  Ibs. 

Value  of  E  in  Ibs. 

1 
2 
4 
8 

•146272 
•413656 
1-169641 

3-308787 

•0064033 
•0180827 
•0512115 
•1448238 

.  

Square roots  of  the 
cubes  of  propor- 
tions of  the  in- 
termediate cir- 
cumferences to 
those  of  the 
table. 


No.  4.     Wetted  half-worn  cords.     Rigidity  proportional 
to  the  square  root  of  the  cube  of  the  circumference. 


Circumference  of 
the  Rope  in  Inches. 

Value  of  D  in  Ibs. 

Value  of  E  in  Ibs. 

1 
2 
4 
8 

•292541 
•827328 
2-339675 
6-616589 

•006401 
•018107 
•051212 
•144822 

THE   RIGIDITY   OF   COEDS.  1 

No.  5.     Tarred  rope.     Rigidity  proportional  to  the  number  of  strands. 


Number  of  Strands. 

Value  of  D  in  Ibs. 

Value  of  E  in  Ibs. 

6 
15 
30 

0-33390 
0-17212 
1-25294 

0-009305 
0-021713 
0-044983 

To  determine  the  constants  D  and  E  for  ropes  whose  circumferences  are 
intermediate  to  those  of  the  tables,  find  the  ratio  of  the  given  circumference 
to  that  nearest  to  it  in  the  tables,  and  seek  this  ratio  or  proportion  in  the  first 
column  of  the  auxiliary  table  to  the  right  of  the  page.  The  corresponding 
number  in  the  second  column  of  this  auxiliary  table  is  a  factor  by  which  the 
values  of  D  and  E  for  the  nearest  circumference  in  the  principal  tables  being 
multiplied,  their  values  for  the  given  circumference  will  be  determined.* 

*  Note  (*)  Ed.  App. 


10 


146  THE  THEOEY  OF  MACHINES. 


T    III. 

THE  THEOEY  OF  MACHINES. 


143.  THE  parts  of  a  machine  are  divisible  into  those  which 
receive  the  operation  of  the  moving  power  immediately,  those 
which  operate  immediately  upon  the  work  to  be  performed, 
and  those  which  communicate  between  the  two,  or  which 
conduct  the  power  or  work  from  the  moving  to  the  working 
points  of  the  machine.     The  first  class  may  be  called  RECEIV- 
ERS, the  second  OPERATORS,  and  the  third  COMMUNICATORS  of 
work. 

THE  TRANSMISSION  OF  WORK  BY  MACHINES. 

144.  The  moving  power  divides  itself  whilst  it  operates  in 
a  machine,  first,  Into  that  which  overcomes  the  prejudicial 
resistances  of  the  machine,  or  those  which  are  opposed  by 
friction  and  other  causes  uselessly  absorbing  the  work  in  its 
transmission.      Secondly,  Into   that   which    accelerates    the 
motion  of  the  various  moving  parts  of  the  machine  ;  so  long 
as  the  work  done  by  the  moving  power  upon  it  exceeds  that 
expended  upon  the  various  resistances  opposed  to  the  motion 
of  the  machine  (Art.  129.).     Thirdly,  Into  that  which  over- 
comes the  useful  resistances,  or  those  which  are  opposed  to 
the  motion  of  the  machine  at  the  working  point  or  points 
by  the  useful  work  which  is  to  be  done  by  it.     Thus,  then, 
the  work  done  by  the  moving  power  upon  the  moving  points 
of  the  machine  (as  distinguished  from  the  working  points) 
divides  itself  in  the  act  of  transmission,  first,  Into  the  work 
expended  uselessly  upon  the  friction  and  other  prejudicial 
resistances  opposed  to  its  transmission.     Secondly,  Into  that 
accumulated  in  the  various  moving  elements  of  the  machine, 
and  reproducible.     Thirdly,  Into  the  useful  work,  or  that 
done  by  the  operators,  whence  results  immediately  the  useful 
products  of  the  machine. 


THE   THEORY    OF   MACHINES. 


145.  The  aggregate  number  of  units  of  useful  works  yielded 
ly  any  machine  at  its  working  points  is  less  than  the  num- 
ber received  upon  the  machine  directly  from  the  moving 
power,  ly  the  number  of  units  expended  ujpon  the  prejudi- 
cial resistances  and  ~by  the  number  of  units  accumulated 
in  the  moving  parts  of  the  machine  whilst  the  work  is  being 
done.* 

For,  by  the  principle  of  vis  viva  (Art.  129.),  if  2U1  repre- 
sent the  number  of  units  of  work  received  upon  the  machine 
immediately  from  the  operation  of  the  moving  power,  %u 
the  whole  number  of  such  units  absorbed  in  overcoming  the 
prejudicial  resistances  opposed  to  the  working  of  the  ma- 
chine, 2U2  the  whole  useful  work  of  the  machine  (or  that 
done  by  its  operators  in  producing  the  useful  effect),  and 

^-Zw^—v*)  one  half  the  aggregate  difference  of  the  vires 


vivae  of  the  various  moving  parts  of  the  machine  at  the 
commencement  and  termination  of  the  period  during  which 
the  work  is  estimated,  then,  by  the  principle  of  vis  VIVA 
(equation  108), 


in  which  vl  and  v9  represent  the  velocities  at  the  commence- 
ment and  termination  of  the  period,  during  which  the  work 
is  estimated,  of  that  moving  element  of  the  machine  whose 
weight  is  w.  Now  one-half  the  aggregate  difference  of  the 
vires  vivse  of  the  moving  elements  represents  the  work  accu- 
mulated in  them  during  the  period  in  repect  to  which  the 
work  is  estimated  (Art.  130.).  Therefore,  &c. 


146.  If  the  same  velocity  of  every  part  of  the  machine  re- 
turn after  any  period  of  time,  or  if  the  motion  ~be  periodical, 
then  is  the  whole  work  received  upon  it  from,  the  moving  power 
during  that  time  exactly  equal  to  the  sum  of  the  useful  work 
done,  and  the  work  expended  upon  the  prejudicial  resistances. 
For  the  velocity  being  in  this  case  the  same  at  the  com- 
mencement and  expiration  of  the  period  during  which  the 
work  is  estimated,  2w(v*— -y2fl)=0,  so  that 

*  Note  (0  Ed.  App. 


148  THE  MODULUS  OF  A  MACHINE. 

(113). 


Therefore,  &c. 

The  converse  of  this  proposition  is  evidently  true. 


147.  If  the  prime  mover  in  a  machine  be  throughout  the 
motion  ^n  equilibrium  with  the  useful  and  the  prejudicial 
resistances,  then  the  motion  of  the  machine  is  uniform. 
For  in  this  case,  by  the  principle  of  virtual  velocities 
(Art.  127.),  2U,— 2U3+2^;  therefore  (equation  112) 
2w(v*— v*)= 0;  whence  it  follows  that  (in  the  case  sup- 
posed) the  velocities  vl  and  v2  of  any  moving  element  of  the 
machine  are  the  same  at  the  commencement  and  termi- 
nation of  any  period  of  the  motion  however  small,  or  that 
the  motion  of  every  such  element  is  a  uniform  motion. 
Therefore,  &c.  » 

The  converse  of  this  proposition  is  evidently  true. 


THE   MODULUS  OF  A  MACHINE    MOVING  WITH  A  UNIFORM   OR 
PERIODICAL  MOTION. 

148.  The  modulus  of  a  machine,  in  the  sense  in  which  the 
term  is  used  in  this  work,  is  the  relation  between  the  work 
constantly  done  upon  it  by  the  moving  power ',  and  that  con- 
stantly yielded  at  the  working  points,  when  it  has  attained 
a  state  of  uniform  motion  if  it  admit  of  such  a  state  of 
motion  ;  or  if  the  nature  of  its  motion  be  periodical,  then 
is  its  modulus  the  relation  between  the  work  done  at  its 
moving  and  at  its  working  points  in  the  interval  of  time 
which  it  occupies  in  passing  from  any  given  velocity  to  the 
same  velocity  again. 

The  modulus  is  thus,  in  respect  to  any  machine,  the  parti- 
cular form  applicable  to  that  machine  of  equation  (113),  and 
being  dependent  for  its  amount  upon  the  amount  of  work  ^u 
expended  upon  the  friction  and  other  prejudicial  resistances 
opposed  to  the  motion  of  the  various  elements  of  the  ma- 
chine, it  measures  in  respect  to  each  such  machine  the  loss 
of  work  due  to  these  causes,  and  therefore  constitutes  a  true 
standard  for  comparing  the  expenditure  of  moving  power  ne- 
cessary to  the  production  of  the  same  effects  by  different  ma- 


THE   MODULUS   OF   A   MACHINE.  14:9 

chines:  it  is  thus  a  measure  of  the  working  qualities  of 
machines.* 

Whilst  the  particular  modulus  of  every^  differently  con- 
structed machine  is  thus  different,  there  is  nevertheless  a 
general  algebraical  type  or  formula  to  which  the  moduli  of 
machines  are  (for  the  most  part  and  with  certain  modifica- 
tions) referable.  That  form  is  the  following, 

U^A  .  U2+B  .  S (114), 

where  U1  is  the  work  done  at  the  moving  point  of  the  ma- 
chine through  the  space  S,  TJ2  the  work  yielded  at  the  work- 
ing points,  and  A  and  B  constants  dependent  for  their  value 
upon  the  construction  of  the  machine  :  that  is  to  say,  upon 
the  dimensions  and  the  combinations  of  its  parts,  their 
weights,  and  the  co-efficients  of  friction  at  their  various  rub- 
bing surfaces. 

It  would  not  be  difficult  to  establish  generally  this  form  of 
the  modulus  under  certain  assumed  conditions.  As  the  mo- 
dulus of  each  particular  machine  must  however,  in  this  work, 
be  discussed  and  determined  independently,  it  will  be  better 
to  refer  the  reader  to  the  particular  moduli  investigated  in 
the  following  pages.  He  will  observe  that  they  are  for  the 
most  part  comprised  under  the  form  above  assumed;  sub- 
ject to  certain  modifications  which  arise  out  of  the  discus- 
sion of  each  individual  case,  and  which  are  treated  at  length. 

149.  There  is,  however,  one  important  exception  to  this 
general  form  of  the  modulus  :  it  occurs  in  the  case  of  ma- 
chines, some  of  whose  parts  move  immersed  in  fluids.  It  is 
only  when  the  resistances  opposed  to  the  motion  of  the  parts 
of  the  machine  upon  one  another  are,  like  those  of  friction, 
proportional  to  the  pressures,  or  when  they  are  constant  re- 
sistances, that  this  form  of  the  modulus  obtains.  If  there^be 
resistances  which,  like  those  of  fluids  in  which  the  moving 
parts  are  immersed  (the  air,  for  instance),  vary  with  the  velo- 
city of  the  motion,  and  these  resistances  be  considerable, 
then  must  other  terms  be  added  to  the  modulus.  This  sub- 
ject will  be  further  discussed  when  the  resistances  of  fluids 
are  treated  of.  It  may  here,  however,  be  observed,  that^if 
the  machine  move  uniformly  subject  to  the  resistance  of^a 
fluid  during  a  given  time  T,  and  the  resistance  of  the  fluid 

*  The  properties  of  the  modulus  of  a  machine  are  here,  for  the  first  time, 
discussed. 


150  THE  MODULUS  OF  A  MACHINE. 

be  supposed  to  vary  as  the  square  of  the  velocity  Y,  then 
will  the  work  expended  on  this  resistance  vary  as  Y2  .  S,  or 
as  Y3  .  T,  since  S= Y  .  T.  If  then  II,  and  IT,  represent  the 
work  done  at  the  moving  and  working  points  during  the 
time  T,  then  does  the  modulus  (equation  114)  assume,  in  this 
case,  the  form 

TJ^A  .  Ua+B  .  Y  .  T-f  C  .  Y3  /T (115). 


THE  MODULUS  OF  A  MACHINE  MOVING  WITH  AN  ACCELERATED 
OK  RETARDED  MOTION. 

150.  In  the  two  last  articles  the  work  IT,,  done  upon  the 
moving  point  or  points  of  the  machine,  has  been  supposed  to 
be  just  that  necessary  to  overcome  the  useful  and  prejudi- 
cial resistances  opposed  to  the  motion  of  the  machine,  either 
continually  or  periodically ;  so  that  all  the  work  may  be  ex- 
pended upon  these  resistances,  and  none  accumulated  in  the 
moving  parts  of  the  machine  as  the  work  proceeds,  or  else 
that  the  accumulated  work  may  return  to  the  same  amount 
from  period  to  period.  Let  us  now  suppose  this  equality  to 
cease,  and  the  work  U1  done  by  the  moving  power  to  exceed 
that  necessary  to  overcome  the  useful  and  prejudicial  resist- 
ances ;  and  to  distinguish  the  work  represented  by  U1  in  the 
one  case  from  that  in  the  other,  let  us  suppose  the  former, 
(that  which  is  in  excess  of  the  resistances)  to  be  represented 
by  U1 ;  also  let  U2  be  the  useful  work  of  the  machine,  done 
through  a  given  space  Sa,  and  which  is  supposed  the  same 
whatever  may  be  the  velocity  of  the  motion  of  the  machine 
whilst  that  space  is  being  described  ;  moreover,  let  Sj  be  the 
space  described  by  the  moving  point,  whilst  the  space  Sa  is 
being  described  by  the  working  point. 

Now  since  Uj  is  the  work  which  must  be  done  at  the 
moving  point  just  to  overcome  the  resistances  opposed  to 
the  motion  of  that  point,  and  U1  is  the  work  actually  done 
upon  that  point  by  the  power,  therefore  U1— U^  is  the  excess 
of  the  work  done  by  the  power  over  that  expended  on  the 
resistances,  and  is  therefore  equal  to  the  work  accumulated 
in  the  machine  (Art.  130.) ;  that  is,  to  one  half  of  the 
increase  of  the  vis  viva  through  the  space  St  (Art.  129.) ;  so 
that,  if  ^  represent  the  velocity  of  any  element  of  the 
machine  (whose  weight  is  w)  when  the  work  U1  began  to  be 
done,  and  v9  its  velocity  when  that  work  has  been  com- 
pleted, then  (Art.  129.), 


THE    VELOCITY    OF   A    MACHINE. 


Now  by  equation  (114)  U^ 
.-.  T?=A  . 


If  instead  of  the  work  I?  done  by  the  power  exceeding  that 
Uj  expended  on  the  resistances  it  had  been  less  than  it,  then, 
instead  of  work  being  accumulated  continually  through  the 
space  S1?  it  would  continually  have  been  lost,  and  we  should 
have  had  the  relation  (Art.  129.), 


so  that  in  this  case,  also, 


The  equation  (116)  applies  therefore  to  the  case  of  a 
retarded  motion  of  the  machine  as  well  as  to  that  of  an 
accelerated  motion,  and  is  the  general  expression  for  the 
modulus  of  a  machine  moving  with  a  variable  motion. 
Whilst  the  co-efficients  A  and  B  of  the  modulus  are  depen- 
dent wholly  upon  the  friction  and  other  direct  resistances  to 
the  motion  of  the  machine,  the  last  term  of  it  is  wholly 
independent  of  all  these  resistances,  its  amount  being  deter- 
mined solely  by  the  velocities  of  the  various  moving  ele- 
ments of  the  machines  and  their  respective  weights. 


THE  TELOCITY    OF  A  MACHINE    MOVING  WITH  A  VARIABLE 

MOTION. 

151.  The  velocities  of  the  different  parts  or  elements  of 
every  machine  are  evidently  connected  with  one  another  by 
certain  invariable  relations,  capable  of  being  expressed  by 
algebraical  formulae,  so  that,  although  these  relations  are 
different  for  different  machines,  they  are  the  same  for  ail 
circumstances  of  the  motion  of  the  same  machine.  In  a 
great  number  of  machines  this  relation  is  expressed  by  a 
constant  ratio.  Let  the  constant  ratio  of  the  velocity  v,  of 
any  element  to  that  V1  of  the  moving  point  in  such  a 


152  THE   VELOCITY   OF   A   MACHINE. 

machine  be  represented  by  X,  so  that  vl=\'Vl,  and  let  v9  and 
V,  be  any  other  values  of  vl  and  Yj  ;  then  -y  =XYa.  Sub- 
stituting these  values  of  v1  and  va  in  equation  (116),  we 
have 


TJ'=A  .  TT,+B  .  SI+(Va'- 


in  which  expression  2i0Xa  represents  the  sum  of  the  weights 
of  all  the  moving  elements  of  the  machine,  each  being  mul- 
tiplied by  the  square  of  the  ratio  X  of  its  velocity  to  that  of 
the  point  where  the  machine  receives  the  operation  of  its 
moving  power.  For  the  same  machine  this  co-efficient  2i0Xa 
is  therefore  a  constant  quantity.  For  different  machines  it 
is  different.  It  is  wholly  independent  of  the  useful  or  pre- 
judicial resistances  opposed  to  the  motion  of  the  machine, 
and  has  its  value  determined  solely  by  the  weights  and 
dimensions  of  the  moving  masses,  and  the  manner  in  which 
they  are  connected  with  one  another  in  the  machine. 
transforming  this  equation  and  reducing,  we  have 


by  which  equation  the  velocity  Y2  of  the  moving  point  of 
the  machine  is  determined,  after  a  given  amount  of  work 
IP  has  been  done  upon  it  by  the  moving  power,  and  a  given 
amount  U2  expended  on  the  useful  resistances  ;  the  velocity 
of  the  moving  point,  when  this  work  began  to  be  done 
being  given  and  represented  by  Yr 

It  is  evident  that  the  motion  of  the  machine  is  more 
equable  as  the  quantity  represented  by  2wX2  is  greater. 
This  quantity,  which  is  the  same  for  the  same  machine  and 
different  for  different  machines,  and  which  distinguishes 
machines  from  one  another  in  respect  to  the  steadiness  of 
their  motion,  independently  of  all  considerations  arising  out 
of  the  nature  of  the  resistances  useful  or  prejudicial  opposed 
to  it,  may  with  propriety  be  called  the  CO-EFFICIENT  OF 
EQUABLE  MOTION.*  The  actual  motion  of  the  machine  is 
more  equable  as  this  co-efficient  and  as  the  co-efficients  A 
and  B  (supposed  positive)  are  greater. 

*  The  co-efficient  of  equable  motion  is  here,  for  the  first  time,  introduced 
into  the  consideration  of  the  theory  of  machines. 


CO-EFFICIENTS   OF   THE   MODULUS.  153 


To    DETERMINE    THE     Co-EFFICIENTS     OF    THE     MODULUS     OF     A 

MACHINE. 

152.  Let  that  relation  first  be  determined  between  the 
moving  pressure  P,  upon  the  machine  and  its  working  pres- 
sure P2,  which  obtains  in  the  state  bordering  upon  motion  by 
the  preponderance  of  P,.  This  relation  will,  in  all  cases 
where  the  constant  resistances  to  the  motion  of  the  machine 
independently  of  P2  are  small  as  compared  with  P2,  be 
found  to  be  represented  by  formulae  of  which  the  following 
is  the  general  type  or  form  :  — 

P,=P3  .*,+*,  .....  (119); 

where  *,  and  $2  represent  certain  functions  of  the  friction 
and  other  prejudicial  resistances  in  the  machine,  of  which 
the  latter  disappears  when  the  resistances  vanish  and  the 
former  does  not;  so  that  if  *j(°>  and  *a(°)  represent  the 
values  of  these  functions  when  the  prejudicial  resistances. 
vanish,  then  $>W=Q  and  *1(°)=  a  given  finite  quantity  - 
dependent  for  its  amount  on  the  composition  of  the  machine... 
Let  P/°>  represent  that  value  of  the  pressure  Pj  which  wxmlftj 
be  in  equilibrium  with  the  given  pressure  P2,  if  there-  ware  • 
no  prejudicial  resistances  opposed  to  the  motion  off  tbie« 
machine.  Then,  by  the  last  equation,  P/°)=P2  .  Qffi. 

But  by  the  principle  of  virtual  velocities  (Art.,12T:),\if 
we  suppose  the  motion  of  the  machine  to  be  unifdrm-j  .  so 
that  Pj  and  P2  are  constantly  in  equilibrium  upon  it,  and  if 
we  represent  by  S,  any  space  described  by  the  point  of 
application  of  P,,  or  the  projection  of  that  space  on  .  the 
direction  of  P,  (Art.  52.),  and  by  S2  the  corresponding 
space  or  projection  of  the  space  described  by  P2,  then 
P/0)  .  S1^P2  .  S2.  Therefore,  dividing  this,  equation  by 
the  last,  we  have 


Multiplying  this  equation  by  equation  (119),  , 


154:  AXES. 

which  is  the  modulus  of  the  machine,  so  that  the  constant 

<f> 
A  in  equation  (114)  is  represented  by  j^,  and  the  constant 

B  by  *r 

The  above  equation  has  been  proved  for  any  value  of  S15 
provided  the  values  of  P,  and  P2  be  constant,  and  the 
motion  of  the  machine  uniform ;  it  evidently  obtains,  there- 
fore, for  an  exceedingly  small  value  of  S15  when  the  motion 
of  the  machine  is  variable. 


GENERAL  CONDITION  OF  THE  STATE  BORDERING-  UPON  MOTION 

IN  A   BODY  ACTED  UPON  BY  PRESSURES    IN   THE    SAME  PLANE, 
AND   MOVEABLE   ABOUT   A   CYLINDRICAL    AxiS. 


153.  If  any  number  of  pressures  P,,  P2,  P3,  <&c.  applied  in 
the  same  plane  to  a  body  moveable  about  a  cylindrical 
axis,  be  in  the  state  bordering  upon  motion,  then  is  the 
direction  of  the  resistance  of  the  axis  inclined  to  its  radius, 
at  the  point  where  it  intersects  the  circumference,  at  an 
angle  equal  to  the  limiting  angle  of  resistance. 

For  let  R  represent  the  resultant  of  P,  P2,  &c.  Then, 
since  these  forces  are  supposed  to  be  upon  the 
11  point  of  causing  the  axis  of  the  body  to  turn 
upon  its  bearings,  their  resultant  would,  if  made 
to  replace  them,  be  also  on  the  point  of  causing 
the  axis  to  turn  on  its  bearings.  Hence  it  fol- 
lows that  the  direction  of  this  resultant  R  cannot 
be  through  the  centre  C  of  the  axis  ;  for  if  it 
f  were,  then  the  axis  would  be  pressed  by  it  in  the 
direction  of  a  radius,  that  is,  perpendicularly 
upon  its  bearings,  and  could  not  be  made  to  turn  upon  them 
by  that  pressure,  or  to  be  upon  the  point  of  turning  upon 
them.  The  direction  of  11  must  then  be  on  one  side  of  C, 
so  as  to  press  the  axis  upon  its  bearings  in  a  direction  RL, 
inclined  to  the  normal  CL  (at  the  point  L,  where  it  inter- 
sects the  circumference  of  the  axis)  at  a  certain  angle  RLC. 
Moreover,  it  is  evident  (Art.  141.),  that  since  this  force  R 
pressing  the  axis  upon  its  bearings  at  L  is  upon  the  point  of 
causing  it  to  -slip  upon  them,  this  inclination  RLC  of  R  to 
the  perpendicular  CL  is  equal  to  the  limiting  angle  of 


THE   WHEEL   AND   AXLE.  155 

resistance  of  the  axis  and  its  bearings.*  Now  the  resistance 
of  the  axis  is  evidently  equal  and  opposite  to  the  resultant 
R  of  all  the  forces  P1?  P2,  &c.  impressed  upon  the  body. 
This  resistance  acts,  therefore,  in  the  direction  LR,  and  is 
inclined  to  CL  at  an  angle  equal  to  the  limiting  angle  of 
resistance.  Therefore,  &c. 


THE  WHEEL  AND  AXLE. 

154.  The  pressures  P,  and  P2  applied  ver- 
tically by  means  of  parallel  cords  to  a 
wheel  and  axle  are  in  the  state  bordering 
upon  motion  by  the  preponderance  of  P15 
it  is  required  to  determine  a  relation 
between  P,  and  P2. 

The  direction  LR  of  the  resistance  of  the  axis  is  on  that 
side  of  the  centre  which  is  towards  P15  and  is  inclined  to  the 
perpendicular  CL  at  the  point  L,  where  it  intersects  the 
axis  at  an  angle  CLR  equal  to  the  limiting  angle  of  resist- 
ance. Let  this  angle  be  represented  by  <p,  and  the  radius 
CL  of  the  axis  by  p ;  also  the  radius  CA  of  the  wheel  by  a,, 
and  that  CB  of  1jie  axle  by  &2 ;  and  let  "W  be  the  weight  of 
the  wheel  and  axle,  whose  centre  of  gravity  is  supposed  to 
be  C.  Now,  the  pressures  Pn  P2,  the  weight  W  of  the 
wheel  and  axle,  and  the  resistance  R  of  the  axis,  are  pres- 
sures in  equilibrium.  Therefore,  by  the  principle  of  the 
equality  of  moments  (Art.  7.),  neglecting  the  rigidity  of  the 
cord,  and  observing  that  the  weight  W  may  be  supposed  to 
act  through  C,  we  have, 

P,  .  CA=P2  .  CB  +  R  .  Cm. 

If,  instead  of  P,  preponderating,  it  had  been  on  the  point 
of  yielding,  or  P2  had  been  in  the  act  of  preponderating, 
then  R  would  have  fallen  on  the  other  side  of  C,  and  we 
should  have  obtained  the  relation  P,  .  CA=P2  .  CB— 
R  .  Cm;  so  that,  generally,  P,  .  CA=P2  .  CB±R  .^Cm; 
the  sign  ±  being  taken  according  as  P,  is  in  the  superior  or 
inferior  state  bordering  upon  motion. 

Now  CA=at,  CB=#2,  Cm= CL  sin.  CLR=p  sin.  <p,  and 

*  The  nide  of  C  on  which  RL  falls  is  manifestly  determined  by  the  direction 
towards  which  the  motion  is  about  to  take  place.  In  this  case  it  is  supposed 
about  to  take  place  to  the  right  of  C.  If  it  had  been  to  the  left,  the  direc- 
tion of  R  would  have  been  on  the  opposite  side  of  C. 


156  THE  WHEEL  AND  AXLE. 


"W;  the  sign  ±  being  taken  according  as  the 
weight  W  of  the  wheel  and  axle  acts  in  the  same  direction 
with  the  pressures  Y1  and  P2,  or  in  the  opposite  direction  ; 
that  is,  according  as  the  pressures  Px  and  Pa  act  vertically 
downwards  (as  shown  in  the  figure)  or  upwards  / 


P  sn.  9, 
/.PX^—  p  sin.  9)=Pa(#a  +  p  sin.  <p)±Wp  sin.  9. 

Now  the  effect  (Art.  142.)  of  the  rigidity  of  the  cord  BPa 
is  the  same  as  though  it  increased  the  tension  upon  that  cord 

(~r\    i    "U*         ~p  \ 
P,+  -        —  —  a):  allowing,  therefore,  for  the 
#a  ' 

rigidity  of  the  cord,  we  have  finally 

(T)-I-"R    P  \ 
P3  +  -  —  -^~  -J  («2  +  p  sin.  <p)±"W  p  sin.  9, 

or  reducing, 


aa^—  p  sm.  9  •          al—  p  sin.  9 

which  is  the  required  relation  between  Px  and  Pa  in  the 
state  bordering  upon  motion. 

p  p 

—  sin.  9  and  —  sin.  9  are  in  all  cases  exceedingly  small  ; 

a,  a,  8  J 

we  may  therefore  omit,  without  materially  affecting  the 
result,  all  terms  involving  powers  of  these  quantities  above 
the  first,  we  shall  thus  obtain  by  reduction 


155.  The  modulus  of  uniform  motion  m  the  wheel  and  axle. 

It  is  evident  from  equation  (122),  that,  in  the  case  of  the 
wheel  and  axle,  the  relation  assumed  in   equation 

/         E  \&a  4-  p  sin.  9 

obtains,  if  we  take  ^=1  1  -f—  I- — — -. — I ; 
1     V         aja,—o  sm.  e  ' 


THE   WHEEL   AND   AXLE. 


157 


and 


al—- p  sin.  9 

^observing  that  *1(°)  represents  the  value  of  $,  when 
the  prejudicial  resistances  vanish  (or  when  9=0  andE=0) 

T_  *  /m         ^ 

we  have  &W=—. 


\— p  sin. 
Therefore  by  equation  (121), 


_/P\  sin.  9  ' 
\aj 


(Si sin- 


1  —    —    sin.  9 


— P  sin.  9 


.(124), 


which  is  the  modulus  of  the  wheel  and  axle. 

Omitting  terms   involving   dimensions   of  —  sin.  9,  and 

Cdl 

—  sin.  9,  and  —  above  the  first,  we  have 

' 


156.  The  modulus  of  variable  motion  in  the  wheel  and  axle. 

If  the  relation  of  P,  and  P2  be  not  that  of  either  state 
bordering  upon  motion,  then  the  motion  will  be  continually 
accelerated  or  continually  retarded,  and  work  will  continu- 
ally accumulate  in  the  moving  parts  of  the  machine,  or  the 
work  already  accumulated  there  will  continually  expend 


158 


THE   WHEEL    AND   AXLE. 


itself  until  tlie  whole  is  exhausted,  and  the  machine  is 
brought  to  rest.  The  general  expression  for  the  modulus  in 
this  state  of  variable  motion  is  (equation  116) 


^ 

^Now  in  this  case  of  the  wheel  and  axle,  if  'Vl  and  Y2  re- 
present the  velocities  of  Pl  at  the 
commencement  and  completion 
of  the  space  S,,  and  «  the  angular 
velocity  of  the  revolution  of  the 
wheel  and  axle  ;  if,  moreover,  the 
pressures  Pt  and  P2  be  supposed 
to  be  supplied  by  weights  sus- 

\  /  pended  from    the   cords ;    then, 

since  the  velocity  of  P2  is  repre- 
sented by  -?—  -,  we  have 
a, 


\  represent  the  moment  of  inertia  of  the  revolving  wheel, 
and  Ia  that  of  the  revolving  axle,  (Art.  75.),  and  if  f\  repre- 
sent the  weight  of  a  unit  of  the  wheel  and  f*2  of  the  axle  ; 
since  2wv*  represents  the  sum  of  the  weights  of  all  the  mov- 
ing elements  of  the  machine,  each  being  multiplied  by  the 
square  of  its  velocity,  and  that  (by  Art.  75.)  a>1I1  represents 
this  sum  in  respect  to  the  wheel,  and  «-V2I2  in  respect  to  the 
axle.  Now,  Y1=a^1, 


a. 


Y3    •!    lCt/l  +r*a*  "T 
Similarly  2^v22=Yaa 


Substituting  in  the  general  expression  (equation  116),  we 
have 


THE  WHEEL  AND  AXLE.  159 

TJ'=AU,+BS1+1(V,"-V,') 


which  is  the  modulus  of  the  machine  in  the  state  of  variable 
motion,  the  co-efficients  A  and  B  being  those  already  deter- 
mined (equation  124),  whilst  the  co-efficient 


ig   the   co.efficient    SwX,  (        tion 

i 

117)  of  equalle  motion.  If  the  wheel  and  axle  be  each  of  them 
a  solid  cylinder,  and  the  thickness  of  the  wheel  be  &„  and  the 
length  of  the  axle  52,  then  (Art.  85.)  Il=faebla\,  I2=i^2&24. 
Now  if  Wl  and  W2  represent  the  weights  of  the  wheel  and 
axle  respectively,  then  Wl=ifal9bl^  W2=tf#22&2M<2  ;  therefore 
M-Ji—  2-W^x2,  M-aIa=i'W'aaa*.  Therefore  the  co-efficient  of 
equable  motion  is  represented  by  the  equation 


or 


(137), 


157.  To  determine  the  velocity  acquired  through  a  given 
space  when  the  r  elation  o£  the  weights  P,  and  P2,  suspended 
from  a  wheel  and  axle,  is  not  that  of  the  state  bordering 
upon  motion* 

Let  Sj  be  the  space  through  which  the  weight  Pa  moves 
whilst  its  velocity  passes  from  Yx  to  V2:  observing  that 

!?=?&,    and  that   Ua=P2Sa=P9^-2,  substituting  in  equa- 

al 

tion  (126),  and  solving  that  question  in  respect  to  Ya,  we 
have 


•  •  .0*); 


*  Note  (w)  Ed.  App. 


160 


THE   PULLEY. 


making  the  same  suppositions  as  in  formula  127,  and  repre- 
senting the  ratio  —  by  ra,  we  have 


THE  PULLEY. 


158.  If  the  radius  of  the  axle  be  taken  equal  to  that  of  the 
wheel,  the  wheel  and  axle  becomes  a  pul- 
ley. Assuming  then  in  equation  122, 

of  the 
state 
when 
the  strings  are  parallel, 


§  ill  ^*      Assuming   then  in    equation 

rfpk  M    Ir  I  «1=«3=«,we  obtain  for  the  relation  < 

I  H  Wk   I  moving  pressures  Yl  and  P2,  in  the 

»  rH  •  C5    \  t)°r(iermg  upon  motion  in  the  pulley, 


..(129); 


and  by  equation  124  for  the  value  of  the  modulus, 


p 

l  +  -sm.  9 
a 


—  ±w)Psin. 

Ob  I 


—?  sn.  9 


.  .  .  (130); 


in  which  the  sign  ±  is  to  be  taken  according  as  the  pressures 
Pt  and  P2  act  downwards,  as  in  the  first  pulley  of  the  pre- 
ceding figure ;  or  upwards,  as  in  the  second.  Omitting 

•p 

dimension  of  -  sin  9,  -  sin.  9,  and  —  above  the  first,  we  have 
a  a  a 

by  equations  (123, 125) 


SYSTEM   OF   ONE   FIXED   ONE   MOVEABLE   PULLET. 


161 


E     2Psin.< 
-  +— 


TT     TT  .         ,     , 

US=TT.    1+-  +—  -     +~ 


Also  observing  that  «,=»„,  and  I,=0,  the  modulus  of  varia- 
ble motion  (equation  126)  becomes 


(133), 

~ij 

and  the  velocity  of  variable  motion  (equations  118,  128)  is 
determined  by  the  equation 


in  which  two  last  equations  the  values  of  A  and  B  are  those 
of  the  modulus  of  equable  motion  (equation  125).. 


SYSTEM  OF  ONE  FIXED  AND  ONE  MQIOJABUE 


159.  In  the  last  article  (equation  131)  it  was 
shown  that  the  relation  between  the  tensions 
P!  and  P2  upon  the  two  parts  of  a  string  pass- 
ing over  a  pulley  and  parallel  to  one  another, 
was,  in  the  state  bordering  upon  motion  by  the 
preponderance  of  P,,  represented  by  an  expres- 
sion of  the  form  Pl=aP9  +  l>,  where  a  and  b  are 
constants  dependent  upon  the  dimensions  of  the 
pulley  and  its  axis,  its  weight,  and  the  rigidity 
of  the  cord,  and  determined  in  terms  of  these 
elements  by  equation  131  ;  and  in  which  ex- 
pression 5  has  a  different  value  according  as  the 
tension  upon  the  cord  passing  over  any  pulley 
acts  in  the  same  direction  with  the  weight  of  that  pulley  (as-. 
in  the  first  pulley  of  the  system  shown  in  the  figure),  or  in. 
the  opposite  direction  (as  in  the  second  pulley)  :  let  these 
different  values  of  5  be  represented  by  I  and  J,.     Now  it  is, 
evident  that  before  the  weight  P2  can  be  raised  by  means  of 
a  system  such  as  that  shown  in  the  figure,  composed  of  one 
fixed  and  one  moveable  pulley,  the  state  of  the  equilibrium 
of  both  pulleys  must  be  that  bordering  upon  motion,  which 
is  described  in  the  preceding  article  ;  since  both  must  be 
upon  the  point  of  turning  upon  their  axes  before  the  weight 
Pa  can  begin  to  be  raised.     If  then  T  and  t  represent  the 
tensions  upon  the  two  parts  of  the  string  which  pass  round, 
the  moveable  pulley,  we  have 


162  SYSTEM  OF   ONE   FIXED   AND   ANY 


and  T= 

Now  the  tensions  T  and  t  together  support  the  weight   P,, 
and  also  the  weight  of  the  moveable  pulley, 


Adding  aT  to  both  sides  of  the  second  of  the  above  equa- 
tions, and  multiplying  both  sides  by  a,  we  have 


Also  multiplying  the  first  equation  by  (1  +  a\ 


Now  if  there  were  no  friction  or  rigidity,  a  would  evi- 
dently become  1  (see  equation  121),  and  <t>°=  *         would 

become-;  the  co-efficients  of  the  modulus  (Art.  148.)  are 


/   a?   \ 
=2(1—  —  ), 

\l  +  ar 


.  -^ 

therefore  A=2(—  —     and  B— 


which  is  the  modulus  of  uniform  motion  to  the  single  move- 
able  pulley.* 

If  this  system  of  two  pulleys  had  been 
arranged  thus,  with  a  different  string  passing 
over  each,  instead  of  with  a  single  string,  as 
shown  in  the  preceding  figure,  then,  represent- 
ing by  t  the  tension  upon  the  second  part  of 
the  string  to  which  P1  is  attached,  and  by  T 
that  upon  the  first  part  of  the  string  to  which 
P2  is  attached,  we  have 


*  The  modulus  may  be  determined  directly  from  equation  (135);  for  it  is 
evident  that  if  Si  and  S2  represent  the  spaces  described  in  the  same  time  by 
PI  and  P2,  then  S1  =  2S2.  Multiplying  both  sides  of  equation  (135)  by  this 
equation,  we  have, 


now  P1S1==U1,  and  P2S2  =  U2,  therefore  &c. 


NUMBER    OF    MOVEABLE    PULLEYS. 


163 


Multiplying  the  last  of  these  equations  by  #,  and  adding  it 
to  the  first,  we  have  Yl(I  +  a)+wa=Ta+b==a*'Pt  +  (l  +  ajb'9 


and  for  the  modulus  (equation  121), 


It  is  evident  that,  since  the  co-efficient  of  the  second  term 
of  the  modulus  of  this  systen  is  less  than  that  of  the  first 
system  (equation  136)  (the  quantities  a  and  b  being  essen- 
tially positive),  a  given  amount  of  work  U2  may  be  done  by 
a  less  expense  of  power  TJ,,  or  a  gived  weight  P2  may  be 
raised  to  a  given  height  with  less  work,  by  means  of  this 
system  than  the  other  ;  an  advantage  which  is  not  due 
entirely  to  the  circumstance  that  the  weight  of  the  move- 
able  pulley  in  this  case  acts  m  favour  of  the  power,  whereas 
in  the  other  it  acts  against  it  ;  and  which  advantage  would 
exist,  in  a  less  degree,  were  the  pulleys  without  weight. 


A  SYSTEM  OF  ONE  FIXED  AND  ANY  J^UMBEK  OF  MOVEABLE 

PULLEYS. 


160.  Let  there  be  a  system  of  n  moveable 
pulleys  and  one  fixed  pulley  combined  as 
shown  in  the  figure,  a  separate  string  passing 
over  each  moveable  pulley  ;  and  let  the  ten- 
sions on  the  two  parts  of  the  string  which 
passes  over  the  first  moveable  pulley  be  re- 
presented by  Tj  and  £„  those  upon  the  two 
parts  of  the  string  which  passes  over  the 
second  by  T2  and  £,,  &c.  Also,  to  simplify 
the  calculation,  let  all  the  pulleys  be  sup- 
posed of  equal  dimensions  and  weights,  and 
the  cords  of  equal  rigidity  ; 


/.T^^  +  fct,  and  Ti+W=T1  +  *1  ; 


/.eliminating,  T  = 


(139). 


Let  the  co-efficients  of  this  equation  be  represented  by  « 
and  /3 


SYSTEM   OF   ONE   FIXED    AND   ANY 


164 


Similarly,  T3=aT8+/3,  Ts=aT4+/3,  T4=aT6+/3,  &c.=&c.. 


Multiplying  these  equations  successively,  beginning  from 
the  second,  by  «,  aa,  a3,  &c.,  a**-1,  adding  them  together,  and 
striking  out  terms  common  to  both  sides  of  the  resulting 
equation,  we  have 


....  -f  a»-l/3; 

or  summing  the   geometrical   progression   in   the  second 
member, 


(140); 


Substituting  for  a  and  (3  their  values  from  equation  (139), 
and  reducing 


\  j 

Whence  observing,  that,  were  there  no  friction,  a  would 

become  unity,  and(--^-j  =  (-)  .   We  have  (equation  121) 
for  the  modulus  of  this  system, 


161.  If  each  cord,  instead  of  having  one  of 
its  extremities  attached  to  a  fixed  obstacle,  had 
been  connected  by  one  extremity  to  a  move- 
able  bar  carrying  the  weight  P2  to  be  raised 
(an  arrangement  which  is  shown  in  the  second 
figure),  then,  adopting  the  same  notation  as 
before,  we  have 


Adding  these  equations  together,  striking  out 
terms  common  to  both  sides,  and  solving  in 
respect  to  T1?  we  have 


NUMBER   OF   MOVEABLE   PULLEYS.  165 

in  which  equation  it  is  to  be  observed,  that  the  symbol  b 
does  not  appear  ;  that  element  of  the  resistance  (which  is 
constant),  affecting  the  tensions  tl  and  £2  equally,  and  there 

fore  eliminating  with  T,  and  T2.  Let  ^  be  represented 
by  «,  then 


—  a^—  -W.    Similarly,  £,=*£,—  -W, 


(143). 


Eliminating  between  these  equations  precisely  as  between 
the  similar  equations  in  the  preceding  case  (equation  140), 
observing  only  that  here  (3  is  represented  by  —  oW,  and  that 
the  equations  (143)  are  n—  1  in  number  intead  of  n,  we  have 


Also  adding  the  preceding  equations  (143)  together,  we  have 

aiW 


Now  the  pressure  P,  is  sustained  by  the  tensions  £„  &,,  &c. 
of  the  different  strings  attached  to  the  bar  which  carries  it. 
Including  P2,  therefore,  the  weight  of  the  bar,  we  have 


Substituting  this  value  of  tn  in  equation  (144), 

n  n^W 

,i=(1_aK_lps+a%+(w_lf_w 

Transposing  and  reducing, 


2 

W( 

2 


166  TACKLE   OF    ANY   NUMBER   OF    SHEAVES. 


«=    «       ,—!+«-; 


r-1 


an 


f\  -      • 

8  ~ 


g-ip.          Wj  n  j 

'""(l  +  a-1)*-!      a  \(l  +  a-l)n-l       )  * 


-  --  al+b.    .  (145). 

a--i 


Whence  observing  that  when  a=l,  ^l  +  a"1)7*—  1}  =  2W—  1; 
we  obtain  for  the  modulus  of  uniform  motion  (equation 
121),  ' 


A  TACKLE  OF  ANY  XUMBER  OF  SHEAVES. 

162.  If  an  number  of  pulleys  (called  in  this  case  sheaves) 
be  made  to  turn  on  as  many  different  centres  in  the  same 
block  A,  and  if  in  another  block  B  there  be  simi- 
larly placed  as  many  others,  the  diameter  of  each 
of  the  last  being  one  half  that  of  a  correspond- 
ing pulley  or  sheave  in  the  first ;  and  if  the  same 
cord  attached  to  the  first  block  be  made  to  pass 
in  succession  over  all  the  sheaves  in  the  two 
blocks,  as  shown  in  the  figure,  it  is  evident  that 
the  parts  of  this  cord  1,  2,  3,  &c.  passing  between 
the  two  blocks,  and  as  many  in  number  as  there 
are  sheaves,  will  be  parallel  to  each  other,  and 
will  divide  between  them  the  pressure  of  a  weight 
P2  suspended  from  the  lower  block  :  moreover, 
that  they  would  divide  this  pressure  between 
them  equally  were  it  not  for  the  friction  of  the 
**  sheaves  upon  their  bearings  and  the  rigidity  of 

the  rope  ;  so  that  in  this  case,  if  there  were  n  sheaves,  the 

tension  upon  each  would  be  -Pa ;  and  a  pressure  ^l  of  that 


n 


TACKLE   OF   ANY   NUMBER   OF   SHEAVES. 


167 


amount  applied  to  the  extremity  of  the  cord  would  be  suffi- 
cient to  maintain  the  equilibrium  of  the  state  bordering  upon 
motion.  Let  T15  T2,  T3,  &c.  represent  the  actual  tensions 
upon  the  strings  in  the  state  bordering  on  motion  by  the  pre- 
ponderance of  Tj,  beginning  from  that  which  passes  from  Pt 
over  the  largest  sheaf;  then 

P^o.T.+S,,  T,=a,TI+Js,  T,=ailT,+5, 
&c.=&c.,Tlr_1=a.T.  +  &.; 

where  a^  #2,  &c.,  5,,  &2,  &c.  represent  certain  constant  co- 
efficients, dependent  upon  the  dimensions  of  the  sheaves  and 
the  rigidity  of  the  rope,  and  determined  by  equation  (131). 
Moreover,  since  the  weight  P2  is  supported  by  the  parallel 
tensions  of  the  different  strings,  we  have 


P  —  T  4-T 

-L   2  -  -Lj-f-  J.a 


4-T 

-f  i 


It  will  be  observed  that  the  above  equations  are  one  more 
in  number  than  the  quantities  T,,  T2,  T3,  &c.  ;  the  latter  may 
therefore  be  eliminated  among  them,  and  we  shall  thus  ob- 
tain a  relation  between  the  weight  P2  to  be  raised  and  that 
Pj  necessary  to  raise  it,  and  from  thence  the 
modulus  of  the  system. 

To  simplify  the  calculation,  and  to  adapt 
it  to  that  form  of  the  tackle  which  is  com- 
monly in  use,  let  us  suppose  another  ar- 
rangement of  the  sheaves.  Instead  of  their 
being  of  different  diameters  and  placed  all 
in  the  same  plane,  as  shown  in  the  last 
figure,  let  them  be  of  equal  diameter  and 
placed  side  by  side,  as  in  the  accompanying 
figure,  which  represents  the  common  tackle. 
The  inconvenience  of  this  last  mode  of  ar- 
rangement is,  that  the  cord  has  to  pass  from 
the  plane  of  a  sheaf  in  one  block  to  the  plane 
of  the  corresponding  sheaf  in  the  other  ob- 
liquely, so  that  the  parts  of  the  cords  be- 
tween the  blocks  are  not  truly  parallel  to 
one  another,  and  the  sum  of  their  tensions  is  not  truly  equal 
to  the  weight  P2  to  be  raised,  but  somewhat  greater  than  it. 
So  long,  however,  as  the  blocks  are  not  very  near  to  one  an- 
other, this  deflection  of  the  cord  is  inconsiderable,  and  the 
error  resulting  from  it  in  the  calculation  may  be  neglected. 
Supposing  the  different  parts  of  the  cord  between  the  blocks 
then  to  be  parallel,  and  the  diameters  of  all  the  sheaves  and 


168  TACKLE   OF   ANY   NUMBER   OF   SHEAVES. 

their  axes  to  be  equal,  also  neglecting  the  influence  of  the 
weight  of  each  sheaf  in  increasing  the  friction  of  its  axis, 
since  these  weights  are  in  this  case  comparatively  small,  the 
co-efficients  «15  a9,  az  will  manifestly  all  be  equal  ;  as  also 


..,  .      ,    23,  . 

&c.=&c.,  T._1=aT.  +  &  f  ' 

also  P2=T1+T2+T3+  .....  +Tn. 

Multiplying  equations  (147)  successively  (beginning  from  the 
second)  by  0,  #2,  «3,  and  an~l  ;  then  adding  them  together, 
striking  out  the  terms  common  to  both  sides,  and  summing 
the  geometric  series  in  the  second  member  (as  in  equation 
140),  we  have 


Cb  -  1 


Adding    equations    (147),    and    observing    that 
.  .  .  .    +Tn=P3,    and  that  P1+T1+Ta+  ....    +Tir_1= 
2—  Tw,  we  have 


Eliminating  Tn  between  this  equation  and  the  last, 


To  determine  the  modulus  let  it  be  observed,  that,  neglect- 
ing friction  and  rigidity,  a  becomes  unity  ;   and  that  for  this 

^(/r  _  ~\\ 
value  of  0,  -     -—  —  becomes   a  vanishing  fraction,   whose 

d    —  1 

value  is  determined  by  a  well  known  method  to  be  -*. 
Hence  (Art.  152.), 

*  Dividing  numerator  and  denominator  of  the  fraction  by  (a—  1)  it  becomes 
a«-i  +  a    v  i  --  :TI»  which  evidently  equals  -  when   a=l.      The  modulus 


may  readily  be  determined  from  equation  (148).  Let  Si  and  S2  represent  the 
spaces  described  by  Px  and  P2  in  any  the  same  time  ;  then,  since  when  the 
blocks  are  made  to  approach  one  another  by  the  distance  Sa,  each  of  the  n  por- 
tions of  the  cord  intercepted  between  the  two  blocks  is  shortened  by  this  dis- 


THE   MODULUS    OF   A   COMPOUND   MACHINE.  169 

nban          I 


Hitherto  no  account  lias  been  taken  of  the  work  expended 
in  raising  the  rope  which  ascends  with  the  ascending  weight. 
The  correction  is,  however,  readily  made.  By  Art.  60.  it 
appears  that  the  work  expended  in  raising  this  rope  (diffe- 
rent parts  of  which  are  raised  different  heights)  is  precisely 
the  same  as  though  the  whole  quantity  thus  raised  had  been 
raised  at  one  lift  through  a  height  equal  to  that  through 
which  its  centre  of  gravity  is  actually  raised.  Now  the  cord 
raised  is  that  which  may  be  conceived  to  lie  between  two 
positions  of  P2  distant  from  one  another  by  the  space  Sa,  so 
that  its  whole  length  is  represented  by  nS^  ;  and  if  j*  repre- 
sent the  weight  of  each  foot  of  it,  its  whole  weight  is  repre- 
sented by  fwS,  :  also  its  centre  of  gravity  is  evidently  raised 
between  the  first  and  second  positions  of  P2  by  the  distance 
£S2  ;  so  that  the  whole  work  expended  in  raising  it  is  repre- 

sented by  JfwSa*  or  by  i^-,   since  S1=7iSa.     Adding  this 

work  expended  in  raising  the  rope  to  that  which  would  be 
necessary  to  raise  the  weight  P2,  if  the  rope  were  without 
weight,  we  obtain* 

TT        a  (a—  1)TT       (  nban 

u-^irr11^  i  ^i- 

which  is  the  MODULUS  of  the  tackle. 


THE  MODULUS  OF  A  COMPOUND  MACHINE. 

163.  Let  the  work  of  a  machine  be  transmitted  from  one 
to  another  of  a  series  of  moving  elements  forming  a  com- 
pound machine,  until  from  the  moving  it  reaches  the  working 
point  of  that  machine.  Let  P  be  the  pressure  under  which 
the  work  is  done  upon  the  moving  point,  or  upon  the  first 
moving  element  of  the  machine  ;  Pj  that  under  which  it  is 

tance  S2,  it  is  evident  that  the  whole  length  of  cord  intercepted  between  the 
two  blocks  is  shortened  by  wS2  ;  but  the  whole  of  this  cord  must  have  passed 
over  the  first  sheaf,  therefore  Si=wS2.  Multiplying  equation  (148)  by  this 
equation,  and  observing  that  TJ^PiS1  and  U2=P2S2,  we  obtain  the  modulus 
as  given  above. 

*  A  correction  for  the  weight  of  the  rope  may  be  similarly  applied  to  the 
modulus  of  each  of  the  other  systems  of  pulleys.  The  effect  of  the  weight  of 
the  rope  in  increasing  the  expenditure  of  work  on  the  friction  of  the  pulleys  if 
neglected  as  unimportant  to  the  result. 


170  MODULUS    OF   A   COMPOUND    MACHINE. 

F*  elded  from  the  first  to  the  second  element  of  the  machine  ; 
t  from  the  second  to  the  third  element,  &c.  ;  and  Pw  the 
pressure  under  which  it  is  yielded  by  the  last  element  upon 
the  useful  product,  or  at  the  working  point  of  the  machine. 
Then,  since  each  element  of  the  compound  machine  is  a  sim- 
ple machine,  the  relation  between  the  pressures  applied  to 
that  element  when  in  the  state  bordering  on  motion  will  be 
found  to  present  itself  under  the  form  of  equation  (119) 
(Art.  152),  in  all  cases  where  the  pressure  under  which  the 
work  upon  each  element  is  done  is  great  as  compared  with 
the  weight  of  that  element  (see  Art.  166.). 

Kepresenting,  therefore,  by  01?  a^  a3  .  .  .  515  52,  J3  .  .  .,  cer- 
tain constants,  which  are  given  in  terms  of  the  forms  and 
dimensions  of  the  several  elements  and  the  prejudicial  resist- 
ances, we  have 


&c.=&c.,  ?_!=«„?.+&„. 

Eliminating  the  n—  1  quantities  P1?  PQ,  Ps  .  .  .,  PB_1,  between 
these  n  equations,  we  obtain  an  equation,  of  the  form, 

P=«P.+&'  .....  (151); 
where  a=^a^az  .  .  .  an,  and 


If  the  only  prejudicial  resistance  to  which  each  element  is 
subjected  be  conceived  to  be  friction,  and  the  limiting  angle 
of  resistance  in  respect  to  each  be  represented  by  <p  ;  then 
considering  each  of  the  quantities  a^  J1?  aa,  52,  as  a  function 
of  9,  expanding  each  by  Maclaurin's  theorem  into  a  series 
ascending  by  powers  of  that  variable,  and  neglecting  terms 
which  involve  powers  of  it  above  the  first,  we  nave 


\  C#P  /  V  WP  / 

where,  a(°\  5/0),  «2(°),  52(°),  represent  the  values  of  a»  b»  a» 

62,  &c.,  when  9=0  and  (  — ^-|     ,          ( -—+- }    ,  &c.  represent 
\  d$  /  \  d$  I 

the  similar  values  of  their  first  differential  co-efficients. 


AXES. 


171 


Let 


Therefore  fl^tf^  (!  +  *,)»  &i=&i(0)  (1  +&)>  09=a,<°>  (1+aN 
£,=^>(l-fj$t),  &c:=&c.;  where  aJ5  ft,  «f,  /3f,  &c.,  each 
involving  the  factor  <p,  are  exceedingly  small.  Substituting 
the  values  of  a19  a25  &c.  in  the  expression  for  <z,  and  neglect- 
ing terms  which  involve  dimensions  of  a1?  aa)  &c.  above  the 
first,  we  have 

a=a<P>  aj®  .  .  .  a^  Jl  +  al+  aa+  a3+  ----  +  aj  ----  (153). 

Now  the  co-e£ficient  of  the  first  term  of  the  modulus  is 
represented  (equation  121)  by  —  -,  a  representing  the  co- 

efficient of  the  first  term  of  equation  (119),  also  substituting 
the  value  of  a  from  equation  (153),  and  observing  that 

a,(o)  ....  ^(o)?  we  have  -.= 


.•.U={l-f.a1  +  a4  +  a14-  ....  +ajUn  +  5.S  ....  (154), 
which  is  the  modulus  of  a  compound  machine  of  n  elements, 
U  representing  the  work  done  at  the  moving  point,  Uw 
that  at  the  working  point,  S  the  space  described  by  the 
moving  point,  and  5  a  constant  determined  by  equation 
(152). 


164.  THE  CONDITIONS  OF  THE  EQUILIBRIUM  OF  ANY  TWO  PRES- 
SURES Pj  AND  P2  APPLIED  IN  THE  SAME  PLANE  TO  A  BODY 
MOVEABLE  ABOUT  A  FIXED  AXIS  OF  GIVEN  DIMENSIONS. 

In.  fig.  1.  the  pressure  P1  and  Pa  are  shown  acting  on  oppo- 
site sides  of  the  axis 
whose  centre  is  C,  and 
in  fig.  2.  upon  the  same 
side.  Let  the  direc- 
tion of  the  resultant 
of  P!  and  Pa  be  repre- 
sented, in  the  first 
case,  by  IR,  and  in 
the  second  by  El.  It 


172 


AXES. 


is  in  the  directions  of  these  lines  that  the  axis  is,  in  the  two 
cases,  pressed  upon  its  bearings.  Suppose  the  relation 
between  P1  and  P2  to  be  such  that  the  body  is,  in  both 
cases,  upon  the  point  of  turning  in  the  direction  in  which 
Pj  acts.  This  relation  obtaining  between  P,  and  P2,  it  is 
evident  that,  if  these  pressures  were  replaced  by  their  re- 
sultant, that  resultant  would  also  be  upon  the  point  of  caus- 
ing the  body  to  turn  in  the  direction  of  Pr  The  direction 
TR  of  the  resultant,  thus  acting  alone  upon  the  body,  lies, 
therefore,  in  the  first  case,  upon  the  same  side  of  the  centre 
C  of  the  axis  as  Pl  does,  and  in  the  second  case  it  lies  upon 
the  opposite  side  ;*  and  in  both  cases,  it  is  inclined  to  the 
radius  CK  at  the  point  K,  where  it  intersects  the  axis  at  an 
angle  CKK,  equal  to  the  limiting  angle  of  resistance  (see 
Art.  153.).  Now,  the  resistance  of  the  axis  acts  evidently  in 
both  cases  in  a  direction  opposite  to  the  resultant  of  P,  and 
P8,  and  is  equal  to  it  ;  let  it  be  represented  by  R.  Upon 
the  directions  of  P15  P2,  and  R,  let  fall  the  perpendiculars 
CA15  CA2,  and  CL,  and  let  them  be  represented  by  al9  a» 
and  \  Then,  by  the  principle  of  the  equality  of  moments, 
since  P1?  P2,  and  R  are  pressures  in  equilibrium, 


If  Pj  had  been  upon  the  point  of  yielding,  or  P2  on  the 
point  of  preponderating,  then  R  would  have  had  its  direction 
(in  both  cases)  on  the  other  side  of  C  ;  so  that  the  last  equa- 
tion would  have  become 


According,  therefore,  as  Pj  is  in  the  superior  or  inferior 
state  bordering  upon  motion, 


And  if  we  assume  X  to  be  taken  with  the  sign  -f  or  —  ,  ac- 
cording as  P!  is  about  to  preponderate  or  to  yield,  then 
generally 


Now,  since  the  resistance  of  the  axis  is  equal  to  the  resultant 
of  Px  and  P2,  if  we  represent  the  angle  PJP3  by  if,  we  have 
(Art.  13.) 

*  The  arrows  in  the  figure  represent,  not  the  directions  of  the  resultants 
but  of  the  resistances  of  the  axis,  which  are  opposite  to  the  resultants. 

f  Care  must  be  taken  to  measure  this  angle,  so  that  PI  and  P2  may  have 


AXES.  173 


Substituting  this  value  of  R  in  the  preceding  equation,  and 
squaring  both  sides, 


(PA-PA)r=x'(P1<+2P1P1  cos. 
transposing  and  dividing  by  P22, 

W(a'~x'}~2(^)  (aA+x'  cos-i)=  ~(a'°~xs)  ; 

solving  this  quadratic  in  respect  to  (  -^  )  , 

\-t  2/ 


P1_fa^a+^a  cos,  i)  ±  4/pA+x2  cos.  *)*— fa8-1*8)  «— Xa) 
P."  a/-*  ; 


cos,  i)  ±  X  4/(<^12  +  2^1^2  cos.  »  +  <%2a)  —  Xa  sin.a  <. 


ISTow  let  the  radius  CK  of  the  axis  be  represented  by  p, 
and  the  limiting  angle  of  resistance  CKR  by  9  ;  therefore 
X=CL=CK  sin.  CKR:=p  sin.  9.  Also  draw  a  straight  line 
from  Aj  to  A2  in  both  figures,  and  let  it  be  represented  by  L  ; 
:.a^—^a^  cos.  AlCAli-\-a^=lu.  Now,  since  the  angles  at 
Ax  and  A2  are  right  angles,  therefore  the  angles  AJA,  and 
AiCAj  are  together  qqual  to  two  right  angles,  or  A^CAj  +  i 
=*;  therefore  AjCAa—  *—  »,  and  cos.  AjCA2=—  cos.  »; 
therefore  L2=«13  +  2<x1«2  cos.  «  +  ^22:  substituting  these  values 
of  L2  and  X  in  the  preceding  equation, 

p  _(X<22  +  p3  cos.  i  sin.  V)  +  p  sin.  9  (L2—  pa  sin.  2»  sin.  *<p)* 

(«1a-P2sin.a9) 
.  P2  .  .  (156). 

The  two  roots  of  the  above  equation  are  given  by  positive 
and  negative  values  of  X,  they  correspond  therefore  (equa- 
tion 155)  to  the  two  states  bordering  upon  motion.  These 
two  values  of  X  are,  moreover,  given  by  positive  and  nega- 
tive values  of  9  ;  assuming  therefore  9  to  be  taken  positively 
or  negatively,  according  as  P,  preponderates  or  yields,  we 
may  replace  the  ambiguous  by  the  positive  sign.  The 

their  directions  both  towards  or  both  from  the  angular  point  I  (as  shown  in  the 
figure),  and  not  one  of  them  towards  that  point  and  the  other  from  it.  Thus, 
in  the  second  figure,  the  inclination  L  of  the  pressures  Pt  and  P2  is  not  the 
angle  AjIPj,  but  the  angle  PJPi.  It  is  of  importance  to  observe  this  distinc- 
tion (see  note  p.  194.). 


AXES. 


relation  above  determined  between  P4  and  P2  evidently 
satisfies  the  conditions  of  equation  (119).  We  obtain  there- 
fore for  the  modulus  (equation  121) 


COS> 


-P  sn. 
U2  .  .  .  (157). 


If  terms  involving  powers  of  |—  1  sin.  <p  above  the  first  be 

neglected,   that  quantity  being  in    all   cases   exceedingly 
small,  we  have 


165.  Jb  determine  the  resultant  R  o/"  any  number  of  pres- 
sures P15  P2,  P3  .  .  .  .,  in  terms  of  those  pressures,  and  the 
cosines  of  their  inclinations  to  one  another. 

Let  «15  a25  ag5  &c.  represent  the  inclinations 
I  AC,  IBC,  &c.  of  the  several  pressures  P,, 
P2,  &c.  to  any  given  axis  CA  in  the  same 
plane;  arid  let  «12,  i13,  iaa,  &c.  represent  the 
inclinations  of  these  pressures  severally  to  one 
another. 
Now  /  AIB^  ZIBC-  ZIAC  (Euc.  I.  32.)  ; 

/.  «12—  «2—  ai5  /.cos.  i12=cos.  ax  cos.  a3-fsin.  ax  sin.  aa. 
Similarly,  cos.  »13=cos.  «x  cos.  «8  +  sin.  ax  sin.  a;? 
cos.  i23=cos.  «2  cos.  a8-fsin.  a2  sin.  «3. 


Now  E2=(P1  cos.  «,+?,  cos.  a2+P3  cos.  «3+  .  .  .  )2  +  (P1 


sin.  ^4-P,  sin.  a2  +  P3  sin.  «3+  .  .  .  )2,  (equation  9,  Art.  11.). 

Squaring  the  two  terms  in  the  second  member,  adding  the 
results,  and  observing  that  cos.  X-j-sin.  X=l, 


Ea=Pa2+P22  +  P32  .  .  .  +2P.P,  (cos.  «,  cos.  aa+sin.  «a  sin.  aa) 
+2P,P8  (cos.  «x  cos.  as-fsin.  ax  sin.  «3)+  .  .  .  .  ; 


AXES. 


175 


+  2PaP8  cos.  IM+  &c 


(160). 


166.  THE  CONDITIONS   OF  THE  EQUILIBRIUM  OF  THREE  PRES 
SURES,  Pj,  P2,  P3,  IN  THE  SAME  PLANE  APPLIED  TO  A  BOD? 

MOVEABLE  ABOUT  A  FIXED  AXIS,  THE  DIRECTION  OF  ONE  OF 
THEM,  P3,  PASSING  THROUGH  THE  CENTRE  OF  THE  AXIS,  AND 
THE  SYSTEM  BEING  IN  THE  STATE  BORDERING  UPON  MOTION 
BY  THE  PREPONDERANCE  OF  P. 


represent  the  inclinations  of  the  directions  of 
the  pressures  P,,  P2,  P3  to  one 
another,  a^  and  $2  the  perpen- 
diculars let  fall  from  the  cen- 
tre of  the  axis  upon  P1  and  P2, 
and  X  the  perpendicular  let 
fall  from  the  same  point  upon 
the  resultant  R  of  P1?  P2,  P3. 
Then,  since  R  is  equal  and 
opposite  to  the  resistance  of 
the  axis  (Art.  153.),  we  have, 
by  the  principle  of  the  equality  of  moments,  P,^—  P2&2— 
XR,  for  P3  passes  through  the  centre  of  the  axis,  and  its 
moment  about  that  point  therefore  vanishes. 

Substituting  the  value  of  Rfrom  equation  (160), 


..  cos.  ilt+ 

2P1P3cos.i13+2P2P3cos.«23.p 

Squaring  both  sides  of  this  equation,  and  transposing, 
P1-(a1«-X-)-2Pl  JP^A-X'CP,  cos..12+P3  cos.  .„)}  = 

-P>22  +  ^2  JP22  +  P32  +  2P2P3  cos.  i,,}  . 
If  this  quadratic  equation  be  solved  in  respect  to  P0  and 


*  In  which  expression  it  is  to  be  understood  that  the  inclination  j12  of  the 
directions  of  any  two  forces  is  taken  on  the  supposition  that  both  the  forces 
act  from  or  both  act  towards  the  point  in  which  they  intersect, 
and  not  one  towards  and  the  other  from  that  point;  so  that  in 
the  case  represented  in  the  accompanying  figure,  the  inclina- 
tion tl2  of  the  two  forces  PI  and  P2  represented  by  the  arrows, 
is  not  the  angle  PiIP2,  but  the  angle  QlPi,  since  IQ  and  IPi  are 
directions  of  these  two  forces,  both  tending  from  their  point 
of  intersection  ;  whilst  the  directions  of  P2I  and  I?!  are  one 
of  them  towards  that  point,  and  the  other  from  it. 


176 


terms  which  involve  powers  of  X  above  the  first  be  omitted, 
we  shall  obtain  the  equation 


2  cos.iw+  O+P3  V+2P2P  A(>s  cos.1,,4-^  cos.«23)  ; 

or  representing  (as  in  Art.  164.)  the  line  which  joins  the 
feet  of  the  perpendiculars,  a^  and  &3  by  L,  and  the  function 
#,  («2  cos.  *!„  +  #!  cos.  i33)  by  M,  and  substituting  for  X  its 
value  p  sin.  9, 


p= 


^Representing  (as  in  Art.  152.)  the  value  of  Pt  when  the 
prejudicial  resistances  vanish,  or  when  9=0,  by  P^0),  we 

haveP^0^  I  —  IP2.     Also  by  the  principle  of  virtual  velo- 
\  d>ll 

cities  P/°>  .  8,=?,  .  S,.     Eliminating  P/0)  between  these 
equations,  we  have  St=  I  —  I  S3.  Multiplying  equation  (161)  by 


Substituting  Ul  for  P^,  U2  for  P2S2,  and  observing  that 


}*....  (162.) 


which  is  the  MODULUS  of  the  system. 

If  P3  be  so  small  as  compared  with  P2  that  in  the  expan- 
sion of  the  binomial  radical  (equation  161),  terms  involving 

p 

powers  of  -p^  above  the  first  may  be  neglected  ;  then, 
" 


*  It  will  be  shown  in  the  appendix,  that  this  equation  is  but  a  particular 
case  of  a  more  general  relation,  embracing  the  conditions  of  the  equilibrium 
of  any  number  of  pressures  applied  to  a  body  moveable  about  a  cylindrical 
axis  of  given  dimensions. 


AXES. 


which  equation  may  be  placed  under  the  form 


"Whence  observing  that  the  direction  of  P3  being  always 
through  the  centre  of  the  axis,  the  point  of  application  of 
that  force  does  not  move,  so  that  the  force  P3  does  not  work 
as  the  body  is  made  to  revolve  by  the  preponderance  of  P,  ;. 
observing,  moreover,  that  in  this  case  the  conditions  of 
equation  (119)  (Art.  152.)  are  satisfied,  we  obtain  for  the? 
modulus 


167.  The  conditions  of  the  equilibrium  of  two  pressures  ~Pl 
and  P2  applied  to  a  body  moveable  about  a  cylindrical  axis, 
taking  into  account  the  weight  of  the  body  and  supposing  it 
to  be  symmetrical  about  its  axis. 

The  body  being  symmetrical  about  its  axis,  its  centre  of 
gravity  is  in  the  centre  of  its  axis,  and  its  weight  produces 
the  same  effect  as  though  it  acted  continually  through  the 
centre  of  its  axis.  In  equation  (161.)  let  then  P2  be  taken  to 
represent  the  weight  W  of  the  body,  and  ija,  iaa  the  inclina- 
tions of  the  pressures  Pt  and  P2  to  the  vertical.  Then 


P,=  (%,+  (P™:±)  |  P,'L'+2P,WM+WV  \  *.  -  (165.) 
\<V          \     a1      I  ( 

Also  by  the  equation  (162)  we  find  for  the  modulus 
U,=U.+ 


And  in  the  case  in  which  P2  is  considerable  as  compared 
with  W,  by  equations  (163,  164). 

12 


178  THE   DIRECTION   OF  THE 


»         /«.\    J  -.    ,     LP     ' 

,  =  1 — I  1 1 H — -am. 

\aj  (       a& 

Ui=  1 1+^-sin.  9  |  U2+  (J^)^nS,  sin.  0  ...  (168.) 


168.  A  MACHINE  TO- WHICH  ABE  APPLIED  ANY  TWO  PRESSURES 
Pt  AND  P2,  AND  WHICH  IS  MOVEABLE  ABOUT  A  CYLINDRICAL 
AXIS,  IS  WORKED  WITH  THE  GREATEST  ECONOMY  OF  POWER 
WHEN  THE  DIRECTIONS  OF  THE  PRESSURES  ARE  PARALLEL, 
AND  WHEN  THEY  ARE  APPLIED  ON  THE  SAME  SIDE  OF  THE 
AXIS,  IF  THE  WEIGHT  OF  THE  MACHINE  ITSELF  BE  SO  SMALL 
THAT  ITS  INFLUENCE  IN  INCREASING  THE  FRICTION  MAY  BE 
NEGLECTED. 

For,  representing  the  weight  of  such  a  machine  by  "W,  and 
neglecting  terms  involving  W  sin.  9,  it  appears  by  equation 
(168)  that  the  modulus  is 


whence  it  follows  that  the  work  U,,  which  must  be  done  at 

the  moving  point  to  yield  a  given  amount  TJ2  at  the  working 

point,  is  less  as  L  is  less. 

Now  L  represents 
the  distance  A^  be- 
tween the  feet  of  the 
perpendiculars  CA,  and 
CA2,  which  distance  is 
evidently  least  when  P, 
and  P2  act  on  the  same 
side  of  the  axis,  as  in 
fig.  2,  and  when  CAt 

and  CA2  are  in  the  same  straight  line  ;  that  is,  when  Pt  and 

P2  are  parallel. 


169.  A  MACHINE  TO  WHICH  ARE  APPLIED  TWO  GIVEN  PRES- 
SURES P!  AND  P2  AND  WHICH  IS  MOVEABLE  ABOUT  A  CYLIN- 
DRICAL AXIS,  IS  WORKED  WITH  THE  GREATEST  ECONOMY  OF 
POWER,  THE  INFLUENCE  OF  THE  WEIGHT  OF  THE  MACHINE 
BEING  TAKEN  INTO  THE  ACCOUNT,  WEEN  THE  TWO  PRESSURES 


GREATEST   ECONOMY   OF   POWER.  179 

ARE  APPLIED  ON  THE  SAME  SIDE  OF  THE  AXIS,  AND  WHEN 
THE  DIRECTION  OF  THE  MOVING  PRESSURE  Px  IS  INCLINED  TO 
THE  VERTICAL  AT  A  CERTAIN  ANGLE  WHICH  MAT  BE  DETER- 
MINED. 

Let  P3  be  taken  to  represent  the  weight  of  the  machine, 
and  let  its  centre  of  gravity  coincide  with  the  centre  of  its 
axis,  then  is  its  modulus  represented  (equation  166.)  by 

^)  +P3'S1V  1  *; 

(Zj  / 

in  which  expression  the  work  Uj,  which  must  be  done  at  the 
moving  point  to  yield  a  given  amount  U2  of  work  at  the 
working  point,  is  shown  to  be  greater  than  that  which  must 
have  been  done  upon  the  machine  to  yield  the  same  amount 
of  work  if  there  had  been  no  friction  by  the  quantity 

r^L  j  u.-L'+aujp.s.Mh  +P.-S.V  1  4 

a^  \q,i 

The  machine  is  worked  then  with  the  greatest  economy  of 
power  to  yield  a  given  amount  of  work,  U2,  when  this  func- 
tion is  a  minimum.  Substituting  for  L2  its  value 

0/4-  2^0,  cos.  «12  +  #22,  and  for  M  its  value  a,  {#2cos.  i18-f 
a^  cos.  »ag}  (see  Art.  166.),  also   for  Sj—  )  its  value  S9.  it  be- 

\Cblf 

comes 


(169.) 

let  us  suppose  that  the  perpendicular  distance  #2  from 
the  centre  of  the  axis  at  which  the  work  is  done,  and  the  in- 
clination ia8  of  its  direction  to  the  vertical,  are  both  given,  as 
also  the  space  S2  through  which  it  is  done,  so  that  the  work 
is  given  in  every  respect  ;  let  also  the  perpendicular  distance 
«j  at  which  the  power  is  applied,  and,  therefore,  the  space  Sx 
though  which  it  is  done,  be  given  ;  and  let  it  be  required  to 
determine  that  inclination  iia  of  the  power  to  the  work  which 
will  under  these  circumstances  give  to  the  above  function  its 
minimum  value,  and  which  is  therefore  consistent  with  the 
most  economical  working  of  the  machine. 

Collecting  all  the  terms  in  the  function  (169.)  which  con- 


180  THE   DIRECTION   OF   THE 

tain  (on  the  above  suppositions)  only  constant  quantities,  and 
representing  their  sum  by  C,  it  becomes 


£-55^  {  2aAU2(U2  cos.  ,12+P3S,  cos.  0  +  0  }  k 

"Now  C  being  essentially  positive,  this  quantity  is  a  mini- 
mum when  201a3TJs(Ua  cos.  '19+P,S?  cos.  i13  is  a  minimum  ;  or, 
observing  that  U2—  P3S3  and  dividing  by  the  constant  factor 
when 

Pa  cos.  i12+P3cos.  »13  is  a  minimum. 

From  the  centre  of  the  axis  C  let  lines  Cp1 
Cps  be  drawn  parallel  to  the  directions  of  the 
pressures  P^  respectively  ;  and  whilst  C/?2 
and  Cj?3  retain  their  positions,  let  the  angle 
^jCPg  or  «18  be  conceived  to  increase  until  Px 
attains  a  position  in  which  the  condition 
P3  cos.  iia  +  P3cos.  »13=a  minimum  is  satisfied. 
Now  ^  ^jOP,=p1Cp1—  #,CP3,  or  ^  IM=IH—  ;23  ; 
Substituting  which  value  of  »23  this  condition 
becomes 

P2  cos.  »12+P8cos.  (»„—»„)  a  minimum, 

or  P2  cos.  '12  +  P3  cos.  »J2  cos.  »23+P3  sin.  <13  sin.  »23  a  minimum, 

or  (Pa+P3  cos.  i23)  cos.  «ia+P8  sin.  i28  sin.  «ia  a  minimum. 

P3  sin.  i29 
Let  now^^  —  ^  -  =  —  =  tan.  7, 


/.  (P2+P3  cos.  »23)  cos.  iM  +  (Pt+P,  cos.  IM)  tan.  7  sin.  i12  is  a  mi- 
nimum, or  dividing  by  the  constant  quantity  (P2+P3  cos.  i23) 
and  multiplying  by  cos.  7, 

cos.  i12  cos.  7  +  sin.  «12  sin.  7=cos.  («12—  7)  is  a  minimum. 


— i 


To  satisfy  the  condition  of  a  minimum,  the  angle  p$p* 
must  therefore  be  increased  until  it  exceeds  180°  by  that 

angle  7,  whose  tangent  is  represented  byp    3p       "     .      To 

Jr  2  -j-  Jr  3  cos.  23 

determine  the  actual  direction  of  Pj  produce  then  pfj  to  <?, 
make  the  angle  qCr  equal  to  7 ;  and  draw  Cm  perpendicular 


GREATEST   ECONOMY   OF   POWER.  181 

to  O,  and  equal  to  the  given  perpendicular  distance  ax  of 
the  direction  of  P,  from  the  centre  of  the  axis.  If  raP*  be 
then  drawn  through  the  point  m  parallel  to  O,  it  will  be  in 
the  required  direction  of  Px ;  so  that  being  applied  in  this 
direction,  the  moving  pressure  Px  will  work  the  machine  with 
a  greater  economy  of  power  than  when  applied  in  any  other 
direction  round  the  axis. 

It  is  evident  that  since  the  value  of  the  angle  «1U  or  pf/pl9 
which  signifies  the  condition  of  the  greatest  economy  of 
power,  or  of  the  least  resistance,  is  essentially  greater  than 
two  right  angles,  Pj  and  P2  must,  TO  SATISFY  THAT  CONDITION, 

BOTH   BE   APPLIED   ON   THE  SAME    SIDE   OF  THE   AXIS.       It  is  then 

a  condition  necessary  to  the  most  economical  working  of  any 
machine  (whatever  may  be  its  weight)  which  is  moveable  about 
a  cylindrical  axis  under  two  given  pressures,  that  THE  MOV- 
ING PRESSURE  SHOULD  BE  APPLIED  ON  THAT  SIDE  OF  THE  AXIS 
OF  THE  MACHINE  ON  WHICH  THE  RESISTANCE  IS  OVERCOME,  OR 

THE  WORK  DONE.  It  is  a  further  condition  of  the  greatest 
economy  of  power  in  such  a  machine,  that  the  direction  in 
which  the  moving  pressure  is  applied  should  he  inclined  to 
the  vertical  at  an  angle  »12,  whose  tangent  is  determined  hy 
equation  (170.). 

When  ia3=0,  or  when  the  work  is  done  in  a  vertical 
direction,  tan.  y— 0;  therefore  »,„=*,  whence  it  follows  that 
the  moving  power  also  must  in  this  case  be  applied  in  a  ver- 
tical direction  and  on  the  same  side  of  the  axis  as  the  work. 

"When  '23=o  or  when  the  work  is  done  horizontally,  tan. 
P 


The  moving  power  must,  therefore,  in  this  case,  be  applied 
on  the  same  side  of  the  axis  as  the  work,  and  at  an  incli- 
nation to  the  horizon  whose  tangent  equals  the  fraction 
obtained  by  dividing  the  weight  of  the  machine  by  the 
working  pressure. 

3* 
Since  the  angle  <12  is  greater  than  *  and  less  than  -^ 

cos.  i,a  is  negative ;  and,  for  a  like  reason,  cos.  »13  is  also  in 
certain  cases  negative.  Whence  it  is  apparent  that  the 
function  (169.)  admits  of  a  minimum  value  under  certain 


182 


THE   PULLET. 


conditions,  not  only  in  respect  to  the  inclination  of  the 
moving  pressure,  but  in  respect  to  the  distance  al  of  its 
direction  from  the  centre  of  the  axis.  If  we  suppose  the 
space  Sj  through  which  the  power  acts  whilst  the  given 
amount  of  work  U2  is  done  to  be  given,  and  substitute  in 
that  function  for  the  product  S2^  its  value  S^,  and  then 
assume  the  differential  of  the  function  in  respect  to  at  to 
vanish,  we  shall  obtain  by  reduction 


U2a 


,  cos.  .13+P32S12 


U22  cos.  ^ 


(in.) 


If  we  proceed  in  like  manner  assuming  the  space  S2  instead 
of  Sj  to  be  constant  and  substituting  in  the  function  (169.) 
for  S  #a  its  value  Saa,,  we  shall  obtain  by  reduction 


=_  _     _ 

P2COS.  I12  +  P3COS.  »13. 

It  is  easily  seen  that  if,  when  the  values  of  iia  and  »23  deter- 
mined by  equation  (170.)  are  substituted  in  these  equations, 
the  resulting  values  of  a:  are  positive,  they  correspond  in 
the  two  cases  to  minimum  values  of  the  function  (169.),  and 
determine  completely  the  conditions  of  the  greatest  economy 
of  power  in  the  machine,  in  respect  to  the  direction  of  the 
moving  pressure  applied  to  it. 


170.     THE     PULLEY,      WHEN     THE      TENSIONS     UPON     THE     TWO 
EXTREMITIES    OF   THE   COED    HAVE   NOT   VERTICAL    DIRECTIONS. 


L 


In  the  case  in  which  the  two  parts  of  the 
string  which  pass  over  a  pulley  are  not 
parallel  to  one  another,  the  relations  estab- 
lished in  Article  158.  no  longer  obtain ; 
and  we  must  have  recourse  to  equation 
(167.)  to  establish  a  relation  between  the 
tensions  upon  them  in  the  state  bordering 
upon  motion.  Calling  "W  the  weight  of 
the  pulley,  a  its  radius,  and  observing 
that  the  effect  of  the  rigidity  of  the  cord, 
in  increasing  the  tension  P1?  is  the  same 
as  though  it  caused  the  tension  P2  to  be- 

"Fi      T) 

come  P2(l  +  — I  +  —  (Art.  142.),  we  have 
\       a  j      a 


THE   PULLET.  183 


D    DL  MW 

-+-rpsiri.9  +  -w   .p  sin.  9 ; 

Gu          Ob  w  it 


or. 


where  L  represents  the  chord  AB  of  the  arc  embraced  by 
the  string,  and  M=&2(cos.  ils  +  cos.  i23),  »13  and  «23  represent- 
ing the  inclinations  of  P1  and  P2  to  the  vertical:  which 
inclinations  are  measured  by  the  angles  P^Pj  and  P2FP3, 
or  their  supplements,  according  as  the  corresponding  pres- 
sures Pj  and  P2  act  downwards,  as  shown  in  the  figure,  or 
upwards  (see  note  to  Article  165.)  ;  so  that  if  both  these 
pressures  act  upwards:  then  the  cosines  of  both  the  angles 
become  negative,  and  the  value  of  M  becomes  negative  ; 
whilst  if  one  only  acts  upwards,  then  one  term  only  of  the 
value  of  M  becomes  negative. 

Substituting  this  value  for  M,  observing  that  L=20  cos.  «, 
where  2»  represents  the  inclination  of  the  two  parts  of  the 
cord  to  one  another  (so  that  2i=»13-H23),  and  omitting  terms 
which  involve  products  of  two  of  the  exceedingly  small 

D   E  o  . 

quantities  —  ,  —  ,  and  -sin.  9  we  have 
a    a'         a 

E    2P  D 


0  cos.* 


Wp(cos.  iia  4-cos.  <23)  sin 
20  cos.  i 


184 


THE   PULLET. 


which  last  equation  is  the  modulus  to  the  pulley,  when  the 
two  parts  of  the  string  are  inclined  to  the  vertical  and  to 
one  another. 


171.  If  both  the  strings  be  inclined  at  equal  angles  to  the 
vertical,  on  opposite  sides  of  it  ;  or  if  i13=uas=»,  so  that  cos. 
i]8  +  cos.  '23=2  cos.  »,  then  equations  (172.)  and  (173.)  become 


P,= 


sn. 


+i,,.  9  .  .  .  (174), 


U  =  |l+-+-pcos.  •  sin.  <e  I  U,+  |-+—  Psin.  9  [s,  .  .  .  (175.) 

/  Ui         Cti      '  )  {  (M  Cb  J 


172.  If  both  parts  of  the  cord  passing  over  a  pulley  be  in 
the  same  horizontal  straight  line,  so  that  the 
pulley  sustains  no  pressure  resulting  from  the 
tension  upon  the  cord,  but  only  bears  its 


weight)   then  <— ^,   and  the   term  involving 

cos.  *  in  each  of  the  above  equations  vanishes.  It  is,  how- 
ever, to  be  observed  that  the  weight  bearing  upon  the  axis 
of  the  pulley  is  in  this  case  the  weight  of  the  pulley 
increased  by  the  weight  of  cord  which  it  is  made  to  support. 
So  that  if  the  length  of  cord  supported  by  the  pulley  be 
represented  by  s,  and  the  weight  of  each  foot  of  cord  by  ^, 
then  is  the  weight  sustained  by  the  axis  of  the  pulley  repre- 
sented by  "W+fxs.  Substituting  this  value  for  "W  in  equa- 
tion (175.),  and  assuming  cos.  <=0,  we  have 


P  sin. 


(176.) 


173.  Let  us  now  suppose  that  there  are  n  equal  pulleys 

sustaining  each  the  same  length 
s  of  cord,  and  let  Uw  represent 
the  work  yielded  by  the  rope 
(through  the  space  S,)  after  it 
has  passed  over  the  n**,  or  last 
pulley  of  the  system,  U,  being 
that  done  upon  it  before  it 
passes  over  the  first  pulley ; 
then  by  Art.  163.,  equations 


THE   PULLEY.  185 

152.  154.  and  176.,  neglecting  terms  involving  powers  of 

"IP    TV   * 

—  .  —  ,  -  sin.  <p  above  the  first,  and  observing  that  #.=#,= 
a    a  a 

TT  "R*  ~\   ( 

&C.=1+-,  ai:=aa:=&c.=—  ,    ^=^=&c.  =  -\ 

Ctr  Of  Gi    \ 

p  sin.  <p  >  ,  we  have 


Representing  the  whole  weight  of  the  cord  sustained  by  the 
pulleys  by  w,  and  observing  that  pns^=w,  we  have 

)?  sin.  9  |  S,  .  .  .  (ITT.) 

In  the  above  equations  it  has  been  supposed,  that  although 
the  direction  of  the  rope  on  either  side  of  each  pulley  is  so 
nearly  horizontal  that  cos.  <  may  be  considered  =  0,  yet  that 
it  does  so  far  lend  itself  over  each  pulley  as  to  cause  the 
surface  of  the  rope  to  adapt  itself  to  the  circumference  of 
the  pulley,  and  thereby  to  produce  the  whole  of  that  resist- 
ance which  is  due  to  the  rigidity  of  the  cord.  If  the  tension 
were  so  great  as  to  cause  the  cord  to  rest  upon  the  pulley 
only  as  a  rigid  rod  or  bar  would,  then  must  we  assume  E=0 
and  D—  0  in  the  preceding  equations. 


174.  If  one  part  of  the  cord  passing  over  a  pulley  have  a 
horizontal,  and  the  other  a  vertical  direction,  as,  for  instance, 
when  it  passes  into  the  shaft  of  a  mine  over  the  sheaf  or 
wheel  which  overhangs  its  mouth  ;  then  one  of  the 

angles  «13  or  «23  (equation  173.)  becomes  -,   and  the 

other  0  or  or,  according  as  the  tension  on  the  ver- 
tical cord  is  downwards  or  upwards,  so  that  cos. 
«13  +  cos.  »a8=±l,  the  sign  ±  being  taken  according 
as  the  tension  upon  the  vertical  cord  is  downwards 
or  upwards.  Moreover,  in  this  case  (Art.  169.) 


«=-  and  cos.  »—  -  ;  therefore  (equation  173.) 
4  4/2 


(W8), 


186 


THE   PULLET. 


174.  The  modulus  of  a  system  of  any  number  of  pulleys,  over 
one  of  which  the  rope  passes  vertically,  and  over  the  rest 
horizontally. 

Let  Uj  repre- 
sent the  work 
done  upon  the 
rope  through 
the  space  Sx  be- 
fore it  passes 
horizontally 
over  the  first 
pulley  of  the 
system,  and  let 
it  pass  horizon- 
tally over  n  such  pulleys;  and  then,  after  having  passed 
over  another  pulley  of  different  dimensions,  let  it  take  a 
vertical  direction,  descending,  for  instance,  into  a  shaft.  Let 
U2  be  the  work  yielded  by  it  through  the  space  Sj  immedi- 
ately that  it  has  assumed  this  vertical  direction :  also  let  u^ 
represent  the  work  done  upon  it  in  the  horizontal  direction 
immediately  before  it  passed  over  this  last  pulley  of  the 
system.  Then,  by  equation  (179.), 


-  + 


sm.  9 


Also,  by  equation  (177.)  representing  the  radius  of  each  of 
the  pulleys  which  carry  the  rope  horizontally  by  #,  the  radius 
of  its  axis  by  p,,  and  its  weight  by  W1?  and  obse 


!  is  here  the  power  and  u:  the  work,  we  have 


observing  that 


sn. 


Eliminating  the  value  of  u^  between  these  equations,  and 

neglecting  powers  above  the  first  in  —  ,  &c.,  we  have 

a 


THE   PIVOT.  187 

.  .  (180.) 


. 

) 


175.  If  the  strings  be  parallel,  and  their  common 
inclination  to  the  vertical  be  represented  by  »,  so 
that  iia  =  iM  =  i;  then,  since  in  this  case  L=2#,  we 
have  (equation  172.),  neglecting  terms  of  more  than 

one  dimension  in  —  and^., 
a        a 


in  which  equation  *  is  to  be  taken  greater  or  less  than  -,  and 

2i 

therefore  the  sign  of  cos.  »  is  to  be  taken  (as  before  explained) 
positively  or  negatively,  according  as  the  tensions  on  the 
cords  act  downwards  or  upwards.  If  the  tensions  are  verti- 
cal, »=0  or  *,  according  as  they  act  upwards  or  downwards, 
so  that  cos.  i  —  ±  1.  The  above  equations  agree  in  this  case, 
as  they  ought  with  equations  (131.)  and  (132.).  If  the  par- 

allel tensions  are  horizontal,  then  i=-,  and  the  terms  inyolr- 
ing  cos.  »  in  the  above  equations  vanish. 


176.  THE  FKICTION  OF  A  PIVOT. 

When  an  axis  rests  upon  its  bearings, 
not  by  its  convex  circumference,  but  by 
its  extremity,  as  shown  in  the  accompany- 
ing figure,  it  is  called  a  pivot.  Let  W 
represent  the  pressure  borne  by  such  a 
pivot  supposed  to  act  in  a  direction  per- 
pendicular to  its  surface,  and  to  press 


188  THE   PIVOT. 


equally  upon  every  part  of  it  ;   also  let  px  represent  the 
radius  of  the  pivot  ;  then  will  *??  represent  the  area  of  the 

W 

pivot,  and  —  -  the  pressure  sustained  by  each  unit  of  that 

•"?! 
area.     And  if  f  represent  the  co-efficient  of  friction  (Art. 

133.),  —  Z  will  represent  the  force  which  must  be  applied 


parallel  to  the  surface  of  the  pivot  to  overcome 
the  friction  of  each  such  unit.  JSTow  let  the  dot- 
ted lines  in  the  accompanying  figure  represent 
an  exceedingly  narrow  ring  of  the  area  of  the  pivot,  and  let 
p  and  p+Ap  represent  the  extreme  radii  of  this  ring;  then 
will  its  area  be  represented  by  *(p  -f-  Ap)2—  tfpa,  or  by  it  j2p(Ap)  -f 
(Ap)aj  ,  or,  since  Ap  is  exceedingly  small  as  compared  with  p, 
by  2tfpAp.  Now  the  friction  upon  each  unit  of  this  area  is 

W/ 

represented  by  —  ^—  ;  therefore  the  whole  friction  upon  the 

*fc 

ring  is  represented  by  —  ^-  .  2tfpAp)  or  by  —  ^-p^p,   and  the 

^Pi3  Pi2 

moment  of  that  friction  about  the  centre  of  the  pivot  by 

—  at  .  paA,  and  the  sums  of  the  moments  of  the  frictions  of 

Pi2 
all  such  rings  composing  the  whole  area  of  the  pivot  by 

V2W  ,     2W  ,     2W   X  , 

2  —  f-  .  P2AP,  or  by  —  ^-2paAP,  or  by  —  f-  I  P24,  or  by 

Pi  Pi  Pi    % 

i3,  or  by  |W/Pl  .......  (183.); 

whence  it  appears  that  the  friction  of  the  pivot  produces  the 
same  effect  to  oppose  the  revolution  of  the  mass  which  rests 
upon  it,  as  though  the  whole  pressure  which  it  sustains  were 
collected  over  a  point  distant  ~by  two-thirds  of  its  radius  from 
its  centre. 

If  6  represent  the  angle  through  which  the  pivot  is  made 
to  revolve,  then  $-p/  will  represent  the  space  described  by 
the  point  last  spoken  of  ;  so  that  the  work  expended  upon 
the  resistance  Wf  acting  there,  would  be  represented  by 
"fWpi/^  which  therefore  represents  the  work  expended  upon 
the  friction  of  the  pivot,  whilst  it  revolves  through  the  angle 


189 

6 ;  so  that  the  work  expended  on  each  complete  revolution 
of  the  pivot  is  represented  by 


ITT.  If  the  pivot  be  hollow,  or  its  surface  be  an  annular 
instead  of  a  continuous  circular  area,  then 
representing  its  internal  radius  by  pa,  and 
observing  that  its  area  is  represented  by 
<7r(pia—  P3a)>  an(i  therefore  the  pressure  upon 

each  unit  of  it  by    .  a_   a  ,  and  the  fric- 


tion  of  each  such  unit  by     .  a       a  ,  we  obtain,  as  before, 

*\Pi  ~Pa  ) 

for  the  friction  of  each  elementary  annulus  the  expression 


—r—  z-.    pApt  and  for  the  sum  of  the  moments  of  the  frictions 

Pi  —  P2 


of  all  the  elements  of  the  pivot  —  —  s£-  /  ^ 


or 


Let  r  represent  the  mean  radius  of  the  pivot,  i.  e.  let 
7l=i(p]  +  pa)  ;  and  let  I  represent  one  half  the  breadth  of  the 
ring,  i.  e.  let  l=^l—  2);  therefore  pl=r+l  and  vj=.r—l. 
These  values  of  pt  and  p2  being  substituted  in  the  above  for- 
mula, it  becomes 


•  or 

° 


(185.); 


whence  it  follows  that  the  friction  of  an  annular  pivot  pro- 
duces the  same  effect  as  though  the  whole  pressure  were  col- 

lected over  a  point  in  it  distant  ly  T*  |  l+i(-j  |  from  it* 

centre,  where  r  represent  its  mean  radius  and  I  one  half  its 
fyreadth.     From  this  it  may  be  shown,  as  before,  that  the 


190  AXES. 


whole  work  expended  upon  each  complete  revolution  of  the 
annular  pivot  is  represented  by  the  formula, 


1Y8.  To  DETERMINE  THE  MODULUS  OF  A  SYSTEM  OF  TWO  PRES- 
SURES APPLIED  TO  A  BODY  MOVEABLE  ABOUT  A  FIXED  AXIS, 
WHEN  THE  POINT  OF  APPLICATION  OF  ONE  OF  THESE  PRES- 
SURES IS  MADE  TO  REVOLVE  WITH  THE  BODY,  THE  PERPEN- 
DICULAR DISTANCE  OF  ITS  DIRECTION  FROM  THE  CENTRE  RE- 
MAINING CONSTANTLY  THE  SAME. 

Let  the  pressures  Pj  and  P2,  instead  of  retaining  constantly 
T,        (as  we  have  hitherto  supposed  them  to  do) 
the  same  relative  positions,  be  now  conceived 
^     continually  to  alter  their  relative  positions  by 
"      the  revolution  of  the  point  of  application  of 
P!  with  the  body,  that  pressure  nevertheless 
retaining  constantly  the  same  perpendicular 
distance  a  from  the  centre  of  the  axis,  whilst 
the  direction  of  P2  and  its  amount  remain 
constantly  the  same. 

It  is  evident  that  as  the  point  A1  thus  continually  alters  its 
position,  the  distance  AjAjj  or  L  will  continually  change,  so 
that  the  value  of  P,  (equation  158.)  will  continually  change. 
]N"ow  the  work  done  under  this  variable  pressure  during  one 
revolution  of  Pa  is  represented  (Art.  51.)  by  the  formula 

^,  if  6  represent  the  angle  AXCA  described  at 

o 

any  time  about  C,  by  the  perpendicular  QA,,  and  therefore 
aj,  the  space  S  described  in  the  same  time  by  the  point  of 
application  A1  of  Pl  (see  Art.  62.). 

Substituting,  therefore,  for  ~P1  its  value  from  equation 
(158.),  we  have 


27T 


AXES. 


Let  now  Ps  be  assumed  a  constant  quantity  ; 

27T  27T 


Now  L=A1A^=  {a'+Sa^  cos. 

2;r  2;r 


o 

27T 


neglecting  powers   of  (-_|_^)   '  above  the  first,  since  in  all 
' 


cases  its  value  is  less  than  unity.  Integrating  this  quantity 
between  the  limits  0  and  2*  the  second  term  disappears,  so 
that 

1      f\,       1 1       IV* 
—JLdd=  -3+- J  2*  nearly; 


.-.PA  .— 


0 

27T 


o 

since  2-au,  is  the  space  through  which  the  point  of  applica- 
tion of  the  constant  pressure  ra  is  made  to  move  in  each  re- 


AXES. 


volution.     Therefore  by  equation  (187),  in  the  case  in  which 
Pa  is  constant, 

U1=U9  j  1+  (i+^a)*P  Bin.  9  }  .....  (188). 


179.  If  the  pressure  P,  be  supplied  by  the  tension  of  a 
rope  winding  upon  a  drum  whose  radius  is  #„  (as  in  the  cap- 
Btan),  then  is  the  effect  of  the  rigidity  of  the  rope  (Art.  142.) 
the  same  as  though  Pa  were  increased  by  it  so  as  to  become 


Now,   assuming  Pa  to  be   constant,  and  observing  that 
Ua=2tfPaaa,  we  have,  by  equation  (1ST), 


Substituting  in  this  equation  the  above  value  for  Pa, 


(  \       aj          «2  )    1  <V^    *^ 

Performing  the  actual  multiplication  of  these  factors,  ob- 
serving that  —  is  exceedingly  small,  and  omitting  the  term 

involving  the  product  of  this  quantity  and — ,  we  have 


o 
Whence  performing  the  integration  as  before,  we  obtain 


^  +  lVp  sin.  9  \ 
al       al  / 


If  this  equation  be  multiplied  by  n,  and  if  instead  of  U,  and 
Ua  representing  the  work  done  during  one  complete  revolu- 
tion, they  be  taken  to  represent  the  work  done  through  n 
such  revolutions,  then 


AXES.  193 

) (189), 


which  is  the  MODULUS. 


180.  If  the  quantity  (~+—  1      be  not  so  small  that  terms 

\  C*a        U/l  I 

of  the  binomial  expansion  involving  powers  of  that  quan- 
tity above  the  first  may  be  neglected,  the  value   of  the 

^»27T 

definite  integral  /Ld&  may  be  determined  as  follows  .:-* 
o 

/»27T  /»27T 

J  (a*  +  Sauces  J  +  a?y<tt—J  {(a,  +  ayi—%a*0 

0  0 


27T  27T 


El(*),  where 


represents  the  complete  elliptic  function  of  the  second  order, 
whose  modulus  is  &*  The  value  of  this  function  is  given 
for  all  values  of  ^  in  a  table  which  will  be  found  at  the  end 
of  this  work. 

Substituting  in  equation  (187), 

.  E,(t)t  .  P,=TT.+ 


*  See  Encyc.  Met.  art.  DBF.  INT.  theorem  2. 

f  An  approximate  value  of  EI(&)  is  given  when  &  is  small  by  the  formula 

=(]  +K-1),  where  K=T.    (See  Encyc.  Met.  art.  DEP.  INT.  equation 


THE  CAPSTAN. 


.  .  .  .  (190). 


THE  CAPSTAN. 

181.  The  capstan,  as  used  on  shipboard,  is  represented  in 

the  accompanying  figure. 
It  consists  of  a  solid  timber 
CO,  pierced  through  the 
greater  part  of  its  length  by 
an  aperture  AD,  which 
receives  the  upper  portion 
of  a  solid  shaft  AB  of  great 
strength,  whose  lower  ex- 
tremity is  prolonged,  and 
strongly  fixed  into  the  tim- 
ber framing  of  the  ship.  The  piece  (DC,  into  the  upper  por- 
tion of  which  are  fitted  the  moveable 
arms  of  the  capstan,  turns  upon  the  shaft 
AB,  resting  its  weight  upon  the  crown  of 
the  shaft,  coiling  the  cable  round  its  cen- 
tral portion  CC,  and  sustaining  the  ten- 
sion of  the  cable  by  the  lateral  resistance 
of  the  shaft.  Thus  the  capstan  combines 
the  resistances  of  the  pivot  and  the  axis, 
so  that  the  whole  resistance  to  its  motion 
is  equal  to  the  sum  of  the  resistances  due  separately  to  the 
axis  and  the  pivot,  and  the  whole  work  expended  in  turning 
it  equal  to  the  whole  work  which  would  be  expended  in 
turning  it  upon  its  pivot  were  there  no  tension  of  the  cable 
upon  it,  added  to  the  whole  work  necessary  to  turn  it  upon 
its  axis  under  the  tension  of  the  cable  were  there  no  friction 
of  the  pivot.  Now,  if  U1  represent  the  work  to  be  done 
upon  the  cable  in  n  complete  revolutions,  the  work  which 
must  be  clone  upon  the  capstan  to  yield  this  work  upon  (he 
cable  is  represented  (equation  189.)  by 


THE   CAPSTAN.  195 

where  ai  represents  the  length  of  the  arm,  and  a2  the  radius 
of  that  portion  of  the  capstan  on  which  the  cable  is  winding. 
Moreover  (Art.  175.),  the  work  due  to  the  friction  of  the 

4 
pivot  in  n  complete  revolutions  is  represented  by  o^Pi/'W. 

On  the  whole,  therefore,  it  appears  that  the  work  Uj 
expended  upon  n  complete  revolutions  of  the  capstan  is 
represented  by  the  formula 


which  is  the  MODULUS  of  the  capstan. 

A  single  pressure  Px  applied  to  a  single  arm  has  been 
supposed  to  give  motion  to  the  capstan ;  in  reality,  a  num- 
ber of  such  pressures  are  applied  to  its  different  arms  when 
it  is  used  to  raise  the  anchor  of  a  ship.  These  pressures, 
however,  have  in  all  cases,— except  in  one  particular  case 
about  to  be  described,— a  single  resultant.  It  is  that  single 
resultant  which  is  to  be  considered  as  represented  by  I*, 
and  the  distance  of  its  point  of  application  from  the  axis 
by  flj,  when  more  than  one  pressure  is  applied  to  move  the 
capstan. 

The  particular  case  spoken  of  above,  in  which  the  pres- 
sures applied  to  move  the  capstan  have  no  resultant,  or  can- 
not be  replaced  by  any  single  pressure,  is  that  in  which 
they  may  be  divided  into  two  sets  of  pressure,  each  set  hav- 
ing a  resultant,  and  in  which  these  two  resultants  are  equal, 
act  in  opposite  directions,  on  opposite  sides  of  the  centre, 
perpendicular  to  the  same  straight  line  passing  through  the 
centre,  and  at  equal  distances  from  it.* 

Suppose  that  they  may  be  thus  compounded  into  the 
equal  pressures  B^  and  K2,  and  let  them  be  replaced  by 
these.  The  capstan  will  then  be  acted  upon  by  four  pres- 
sures,— the  tension  P2  of  the  cable,  the  resistance  K  of  the 
shaft  or  axis,  and  the  pressures  Kx  and  Ka.  Now  these  pres- 
sures are  in  equilibrium.  If  moved,  therefore,  parallel  to 
their  present  directions,  so  as  to  be  applied  to  a  single  point, 

*  Two  equal  pressures  thus  placed  constitute  a  STATICAL  COUPLE.  The  pro- 
perties of  such  couples  have  been  fully  discussed  by  M.  Poinsot,  and  by  Mr. 
Pritchard  in  his  Treatise  on  Statical  Couples ;  some  account  of  them  will  be 
found  in  the  Appendix  to  this  work. 


196  THE   CAPSTAN. 

they  would  be  in  equilibrium  about  that  point  (Art.  8.), 
But  when  so  removed,  Rj  and  Ra  will  act  in  the  same 
straight  line  and  in  opposite  directions.  Moreover,  they 
are  equal  to  one  another ;  Rx  and  R2  will  therefore  sepa- 
rately be  in  equilibrium  with  one  another  when  applied  to  that 
point ;  and  therefore  P2  and  R  will  separately  be  in  equili- 
brium ;  whence  it  follows,  that  R  is  equal  to  P2  or  the  whole 
pressure  upon  the  axis,  equal  in  this  case  to  the  whole  tension 
P2  upon  the  cable.  So  that  the  friction  of  the  axis  is  repre- 
sented in  every  position  of  the  capstan  by  P2  tan.  9  (tan.  9 
being  equal  to  the  co-efficient  of  friction  (Art.  138.)),  and 
the  work  expended  on  the  friction  of  the  axis,  whilst  the 
capstan  revolves  through  the  angle  d  by  P2pd  tan.  9,  or  by 

I— J  tan.  9,  or  by  U2  |— j  tan.  9 ;  so  that,  on  the  whole, 

introducing  the  correction  for  rigidity  and  for  the  friction  of 
the  pivot,  the  modulus  (equation  191)  becomes  in  this  case 


j  D+|p,/W  }  .  .  .  .  (192). 


This  is  manifestly  the  least  possible  value  of  the  modulus, 
being  very  nearly  that  given  (equation  191)  by  the  value 
infinity  of  ar* 

Tims,  then,  it  appears  generally  from  equation  (191),  that 
the  loss  by  friction  is  less  as  al  is  greater,  or  as  Px  is  applied 
at  a  greater  distance  from  the  axis  ;  but  that  it  is  least  of  all 
when  the  pressures  are  so  distributed  round  the  capstan  as 
to  be  reducible  to  a  COUPLE,  that  case  corresponding  to  the 
value  infinity  of  at.  This  case,  in  which  the  moving  pres- 
sures upon  the  capstan  are  reducible  to  a  couple,  manifestly 
occurs  when  they  are  arranged  round  it  in  any  number  of 
pairs,  the  two  pressures  of  each  pair  being  equal  to  one  an- 
other, acting  on  opposite  sides  of  the  centre,  and  perpendi- 
cular to  the  same  line  passing  through  it.  This  symmetrical 
distribution  of  the  pressures  about  the  axis  of  the  capstan  is 
therefore  the  most  favourable  to  the  working  of  it,  as  well 
as  to  the  stability  of  the  shaft  which  sustains  the  pressure 
upon  it. 

*  £  being  exceedingly  small,  tan.  #  is  very  nearly  equal  to  sin.  <f>> 


AXES.  197 


182.  THE  MODULUS  OF  A  SYSTEM  OF  THREE  PRESSURES  APPLIED 
TO  A  BODY  MOVEABLE  ABOUT  A  CYLINDRICAL  AXIS,  TWO  OF 

THESE  PRESSURES  BEING  GIVEN  IN  DIRECTION  AND  PARAL- 
LEL TO  ONE  ANOTHER,  AND  THE  DIRECTION  OF  THE  THIRD 
CONTINUALLY  REVOLVING  ABOUT  THE  AXIS  AT  THE  SAME 
PERPENDICULAR  DISTANCE  FROM  IT. 

Let  P3  and  P3  represent  the  parallel  pressures  of  the  sys- 
V^  tern,  and  'Pl  the    revolving  pressure. 

/f'\  From  the  centre  of  the  axis  C,  let  fall 
/'/  v  the  perpendiculars  CA^  CA2,  CA3  upon 
H/d  the  directions  of  the  pressures,  and  let 

L-U.  &  represent  the  inclination  of  CAX  to 

C  A3  -at  any  period  of  the  revolution  of 
t*».  ai  H         '  PI-      Let  Px   be    the  preponderating 

pressure,  and  let  P2  act  to  turn  the 

system  in  the  same  direction  as  P1?  and  P3  in  the  opposite 
direction  ;  also  let  R  represent  the  resultant  of  P2  and  P3, 
and  r  the  perpendicular  distance  CA  of  its  direction  from  C. 
Suppose  the  pressures  P2  and  P8  to  be  replaced  by  R  ;  the 
conditions  of  the  equilibrium  of  Pt  throughout  its  revolu- 
tion, and  therefore  the  work  of  Px  will  remain  unaltered  by 
this  change,  and  the  system  will  now  be  a  system  of  two 
pressures  P,  and  R  instead  of  three  ;  of  which  pressures  R 
is  given  in  direction.  The  modulus  of  this  system  is  there- 
fore represented  (equation  187)  by  the  formula 


(193); 


where  Ur  represents  the  work  of  R,  and  L  represents  the  dis- 
tance AA,  between  the  feet  of  the  perpendiculars  r  and  al9 
so  that  V—a^—Zay  cosJ-^-rt=(a—r  cos.  d)2+7*a  sin.2d  ; 


/.  R2La=(R^-R^  cos. 
Now,  R= 


[Now  if  'the  relations  of  a,  to  #3  are  such  that 
|  (P3  +  PaX_(P3a3-P2<)  cos.  6  |  2>(P3«3-P2<)sin.94 
then  the  value  of  R2L2  will  be  represented  by  the  sum  of  the 


198  AXES. 

squares  of  two  quantities  the  first  of  which  is  greater  than 
the  second.  ED.]  Therefore,  extracting  the  square  root  by 
Poncelet's  theorem,  (see  Appendix  B.) 

RL=a}(PI  +  P1)al_(P1«l_ps09)  cos.  &}  +/3(P3a9-P2A2)  sin.  6 
very  nearly  ;  or, 

-/3smJ).  .  .  .(194). 


\  - 


cos.    -    sn. 
o 
e  6 


0  0 

(a  cos.  d—  /3  sin.  fy#.   .  .  .  (195). 

If  P2  and  P3  be  constant,  the  integral  in  the  second  member 
of  this    equation  becomes  (~Paa3—  P2&2)  («  sin.  6  +  /3  cos.  0)  ; 

,  .        .  -D         P.a.d—  P,a,d     TJ3—  U3 

whence  observing  that  Psa3—  P2<^2=    8  3   ^  —  2-^-=—  L.-  —  ?; 

also,  that  IJr==dEr==^Ptfl^-r-*P1at=IJ,—  TJ,,  and  substituting 
in  equation  (193),  we  have 

U.^TJ.-U^+p  sin.  9  I  «  (-•+-')  - 
(     \a9      az  I 

(5«2«\(«  sin.  4+/S  cos.  6)  \  .  .  .  .  (196)  ; 

\        df        I 

for  complete  revolution  making  0=2*,  we  have 

n-u.-u.+,-. 

reducing, 


which  is  the  modulus  of  the  system  where  a  and  j3  are  to  be 
de'ermined,  as  in  Note  B,  (Appendix.) 


THE   CHINESE   CAPSTAN.  199 

183.  If  the  pressure  P3  be  supplied  by  the  tension  of  a  cord 
which  winds  upon  a  cylinder  or  drum  at  the  point  A3,  then 
allowance  must  be  made  for  the  rigidity  of  the  cord,  and  a 
correction  introduced  into  the  preceding  equation  for  that 
purpose.  To  make  this  correction  let  it  be  observed 
(Art.  142.)  that  the  effect  of  the  rigidity ^ of  the  cord  at  A8  is 
the  same  as  though  it  increased  the  tension  there  from 

D 


or  (multiplying  both  sides  of  this  inequality  by  a,,  and  inte- 
grating in  respect  to  d,)  as  though  it  increased 

27T  27T  27T 

dA  to    l-f- 


or,U3to(l+-)U 

\       &zi 

Thus  the  effect  of  the  rigidity  of  the  rope  to  which  P3  is  ap- 
plied upon  the  work  U  a  of  that  force  is  to  increase  it  to 

(l  +  —  )  U3  +  2*-D.     Substituting  this  value  for  U3  in  equa- 

tion (197),  and  neglecting  terms  which  involve  products  of 

,,  ,.     ..  ,,  ....      E    P  sin.  <p  p  sin.  9   __j  pv 

the  exceedingly  small   quantities—,^  -  ,-  --  ,andlJ, 

we  have 


To  determine  the  modulus  for  n  revolutions  we  must  sub- 
stitute in  this  expression  w*  for  if. 


THE  CHINESE  CAPSTAN. 
184.   This  capstan  is  represented  in  the  accompanying 


200  THE   CHINESE   CAPSTAN. 

figure  under  an  exceed- 
ingly portable  and  con- 
venientform.*  The  axle 
or  drum  of  the  capstan 
is  composed  of  two  parts 
of  different  diameters. 
One  extremity  of  the 
cord  is  coiled  upon  one 
of  these,  and  the  other,  in  an  oposite  direction,  upon  the 
other  ;  so  that  when  the  axle  is  turned,  and  the  cord  is 
wound  upon  one  of  these  two  parts  of  the  drum,  it  is,  at  the 
same  time,  wound  off  the  other,  and  the  intervening  cord  is 
shortened  or  lengthened,  at  each  revolution,  by  as  much  as 
the  circumference  of  the  one  cylinder  exceeds  that  of  the 
other.  In  thus  passing  from  one  part  of  the  drum  to  the 
other,  the  cord  is  made  to  pass  round  a  moveable  pulley 
which  sustains  the  pressure  to  be  overcome. 

To  determine  the  modulus  of  this  machine,  let  u^  and  u3 
represent  the  work  done  upon  the  two  parts  of  the  cord 
respectively,  whilst  the  work  U^  is  done  at  the  moving  point 
of  the  machine,  and  U2  yielded  at  its  working  point. 

Then,  since  in  this  case  we  have  a  body  moveable  about  a 
cylindrical  axis,  and  acted  upon  by  three  pressures,  two  of 
which  are  parallel  and  constant,  viz.  the  tensions  of  the  two 
parts  of  the  cord  ;  and  the  point  of  application  of  the  third 
is  made  to  revolve  about  the  axis,  remaining  always  at  the 
same  perpendicular  distance  from  it  ;  it  follows  (by  equation 
198),  that,  for  n  revolutions  of  the  axis, 


(199); 
where 


in.9(---)  Land 
\aa     %najt]  \ 


#2  and  as  representing  the  radii  of  the  two  parts  of  the  drum, 
#!  the  constant  distance  at  which  the  power  is  applied,  and  p 
the  radius  of  the  axis. 

*  A  figure  of  the  capstan  with  a  double  axle  was  seen  by  Dr.  0.  Gregory 
among  some  Chinese  drawings  more  than  a  century  old.  It  appears  to  have 
been  invented  under  the  particular  form  shown  in  the  above  figure  by  Mr.  G. 
Eckhardt  and  by  Mr.  M'Lean  of  Philadelphia.  (See  Professor  Robinson's  Mech. 
Phil.  vol.  ii.  p.  255.) 


THE   CHINESE   CAPSTAN. 


201 


Also,  since  the  two  parts  of  the  cord  pass  over  a  pulley, 

and  the  pulley  is  made 
to  revolve  under  the  ten- 
sions of  the  two  parts  of 
the  cord,  ?£3  being  the 
work  of  that  tension 
which  preponderates,  we 
have  (by  equation  181), 
if  S  represents  the  length 
of  cord  which  passes 
over  the  pulley, 


where 


E 


a 


and 


7 


D 


.       ) 
m'*    '    • 


a  representing  the  radius  of  the  pulley,  Pl  the 
axis,  W  its  weight,  and  »  the  inclination  of  the  direction  of 
the  tensions  of  the  two  parts  of  the  cord  to  the  vertical,  the 
axis  of  the  pulley  being  supposed  horizontal,  and  the  two 

parts  of  the  cord  parallel.     Now  t3=    U*    ,  L  =-^—  .    Sub- 

2n*0.         2wr0a 
stituting  these  values,  and  multiplying  by  2n^a^  we  have 


''    , 

te-v- • .  •-•• 


(200). 


Since  the  tensions  4  and  ts  of  the  two  parts  of  the  cord, 
and  the  pressure  P2  overcome  by  the  machine,  are  pressures 
applied  to  the  pulley  and  in  equilibrium,  and  that  the  points 
of  application  of  £?  and  P2  are  made  to  move  in  directions 
opposite  to  those  in  which  those  pressures  act,  whilst  the 
point  of  application  of  t5  is  made  to  move  in  the  same  direc- 
tion ;  therefore  (Art.  59.), 


Eliminating  uz  and 
(200),  we  have 


between  this  equation  and  equation 


202  THE    HORSE    CAPSTAN. 

Substituting  these  values  in  equation  (199),  and  reducing, 


a*      \ 
Substituting  their  values  for  A,  A,,  B,  B1?  neglecting  terms 

•  "T^ 

involving  more  than  one  dimension  of  - — — ,  — ,  &c.  and 
reducing,  we  obtain  for  the  MODULUS  of  the  machine, 


E  ,      .        (2a      /       0v     ft     ) 

— hpsm.  <p  \ (1 - V 

!     <h (  a3      \       aj  %rw.f  ) 

.,     a«    E     2p.    . 

1— — -| (--^  sin.  9 


-D 


2p1sin. 


..  (201). 


From  which  expression  it  is  apparent  that  when  the  radii  #a 
and  $3  of  the  double  axle  are  nearly  equal,  a  great  sacrifice 
of  power  is  made,  in  the  use  of  this  machine,  by  reason  of 
the  rigidity  of  the  cord. 


THE  HOUSE  CAPSTAN,  OR  THE  WHIM  GIN. 

185.  The  whim  is  a  form  of  the  capstan,  used  in  the  first 
operations  of  mining,  for  raising  materials  from  the  shaft  and 
levels  by  the  power  of  horses,  before  the  quantity  excavated 
is  so  great  as  to  require  the  application  of  steam  power,  or 
before  the  valuable  produce  of  the  mine  is  sufficient  to  give 
a  return  upon  the  expenditure  of  capital  necessary  to  the 
erection  of  a  steam  engine.  The  construction  of  this  machine 
will  be  sufficiently  understood  from  the  accompanying  figure. 
It  will  bo  observed  that  there  are  two  ropes  wound  upon  the 
drum  in  opposite  directions,  and  which  traverse  the  space 


THE   HORSE   CAPSTAN.  203 

between  the  capstan 
and  the  mouth  of  the 
shaft.  One  of  these 
carries  at  its  extrem- 
ity the  descending 
(empty)  bucket,  and  is 
continually  in  the  act 
of  winding  off  the  drum  of  the  capstan  as  it  revolves ;  whilst 
the  other,  from  whose  extremity  is  suspended  the  ascending 
(loaded)  bucket,  continually  winds  on  the  drum.  The  pres- 
sure exerted  by  the  horses  is  that  necessary  to  overcome  the 
friction  of  the  different  bearings,  and  the  other  prejudicial 
resistances,  and  to  balance  the  difference  between  the  weight 
of  the  ascending  load,  bucket,  and  rope,  and  that  of  the 
descending  bucket  and  rope.  The  rope,  in  passing  from  the 
capstan  to  the  shaft,  traverses  (sometimes  for  a  considerable 
distance)  a  series  of  sheaves  or  pulleys,  such  as  those  shown 
in  the  accompanying  figure. 

Let  now  #2  represent  the  radius  of  the  drum  on  which  the 
rope  is  made  to  wind,  and  n  the  number  of  resolutions 
which  it  must  make  to  wind  up  the  whole  cord ;  also  let  f* 
represent  the  weight  of  each  foot  of  cord,  and  d  the  angle 
which  the  capstan  has  described  between  the  time  when  the 
ascending  bucket  has  attained  any  given  position  in  the 
shaft  and  that  when  it  left  the  bottom ;  then  does  a£  repre- 
sent the  length  of  the  ascending  rope  wound  on  the  drum, 
and  therefore  of  the  descending  rope  wound  off  it.  Also, 
let  "W  represent  the  whole  weight  of  the  rope ;  then  does 
W— pajb  represent  the  weight  of  the  ascending  rope,  and 
f*0ad  that  of  the  descending  rope,  both  of  which  hang  sus- 
pended in  the  shaft.  Let  P2  represent  the  load  raised  at 
each  lift  in  the  bucket,  and  w  the  weight  of  the  bucket ; 
then  is  the  tension  upon  the  ascending  rope  at  the  mouth  of 
the  shaft  represented  by  W— M-a^+x^+w,  and  that  upon 
the  descending  rope  by  pa£+w. 

Let,  moreover,  p3  and  j!?2  represent  the  tensions  upon  these 
ropes  after  they  have  passed  from  the  mouth  of  the  shaft, 
over  the  intervening  pulleys,  to  the  circumference  of  the 
capstan. 

Now,  since  the  tension  upon  the  ascending  rope,  which  is 
"V\T— (x&2d-fPa-{-w  at  the  mouth  of  the  shaft,  is  increased  to 
ps  at  the  capstan,  and  that  the  tension  upon  the  descending 
rope,  which  is  p^  at  the  capstan,  is  increased  to  pa£+w  at 
the  mouth  of  the  shaft,  if  we  represent  by  (1  -f-  A)  and  B  the 
constants  which  enter  into  equation  180  (Art.  174.),  we  have. 


204  THE  HORSE  CAPSTAN. 

by  that  equation  (observing  that  U1=P1S1  and  Ua=P2Sj«  so 
that  S,  disappears  from  both  sides  of  it), 

^3=(l+A)(W+P2+w-^)  +  B,  .  .  .  .(202), 
and  f*a1«+w=(l+A)j>1+B  ....  (203). 


Multiplying  the  former  of  the  above  equations  by 
adding  them,  transposing,  dividing  by  (1  -h  A),  and  neglect- 
ing terms  of  more  than  one  dimension  in  A  and  B, 


Now  Ur  in  equation  (193)  represents  the  work  of  J;he 
resultant  of  pz  and  pz  during  n  revolutions  of  the  capstan,  it 
therefore  equals  the  difference  between  the  work  of  p3  and 
that  of  pz  (see  p.  198). 


2»7T  2tt7T 

9—p,)  d6 


2717T 


o 
{  (1  +  A)  (W  +  P2)  4-  2  Aw  +  2B  }  (2  wa.)  —  f 


observing  that  2nrra2=:S2,  and  that  P2S2=U2. 

Now,  let  it  be  observed  that  the  pressures  applied  to  the 
capstan  are  three  in  number  ;  two  of  them,  ps  and  p^  being 
parallel  and  acting  at  equal  distances  a.2  from  its  axis  ;  and 
the  third,  P1?  being  made  to  revolve  at  the  constant  distance 
at  from  the  axis  (P,  representing  the  pressure  of  the  horses, 
or  the  resultant  of  the  pressures  of  the  horses,  if  there  be 
more  than  one,  and  a^  the  distance  at  which  it  is  applied)  ; 
BO  that  equation  193  (Art.  182.)  obtains  in  respect  to  the 
pressures  P.,  p^  pz  ;  a3  being  assumed  equal  to  av 

Substituting^  and^?8  for  P2  and  P8  in  equation  (194), 

a«i(^8-j-^a)_  a9(p3—  p^)  (a  cos.  0—  p  sin.  0)  ; 


0  0 

(a  cos.  6— (3  sin.  6)  dO. 


THE   HORSE   CAPSTAN. 


205 


Now,  the  terms  of  equation  (180),  represented  in  the  above 
equations  by  A  and  B,  are  all  of  one  dimension  in  the  exceed- 
ingly small  quantities  D,  E,  sin.  <p.  If,  therefore,  the  values 
of  p^  and  pz  given  by  these  equations  be  substituted  in  the 

2«7T 

value  of  LSL$/EL<#  (equation  193),  then  all  the  terms 
av    t/ 

o 

of  that  expression  which  involve  the  quantities  A  and  B  will 
be  at  least  of  two  dimensions  in  D,  E,  sin.  9,  and  may  be  ne- 
glected. Neglecting,  therefore,  the  values  of  A  and  B  in 
equations  (202,  203),  we  obtain 


Similarly, 


f2  /  (Pi— jpyaeos. 
«/ 

0 


c$  P, 


representing  by  S2  the 
space  described  by  the 
load,  and  by  U3  the 
useful  work  done  upon 
it,  during  n  revolu- 
tions of  the  capstan. 


2n7T 


-  )8  sin.  0)dO  =  a 


(a  cos.  0—P  sin.  6)de= 


2nir 

a  cos.  6— (3  sin. 


a  cos.  ^—^  sin. 


206  THE   HORSE   CAPSTAN. 

Now  Aa  cos.  0—(3  sin.  6)dO=(3,  and  Aa  cos.  0— (3  sin. 


«,jp,—  />,)(*  cos.  0—  £  sin.  fyZ0==00, 


=/3a  2^+^2(W-2fxS2)  ;  observing  that  P2 


2717T 


Substituting  this  value,  and  also  that  of  Ur  (equation  204) 
in  equation  (193),  and  assuming 


and  Ca= 
we  have 


00  0 

/•  /*  /* 

*  For   I  0cos.  0f?0=0sin.  0—  /  sin.  0d0*=0sin.0— vers.  0;  also  I  0  sin.  0d0 

00  0 

0 

=—0  cos.  0  +  /cos.  6dO*=—6  cos.  0-f-sin.  0.    Now,  substituting  2w7r  for  0, 


0 
these  integrals  become  respectively  0  and  —  2mr. 

*  Church's  Diff.  and  Int.  Cal.    Art.  140. 


THE   FRICTION    OF   CORDS.  207 


+        i-     g  _ 
•  a,       I    '  , 

which  is  the  MODULUS  of  the  machine,  all  the  various  ele- 
ments, whence  a  sacrifice  of  power  may  arise  in  the  working 
of  it,  being  taken  into  account. 


THE  FRICTION  OF  CORDS. 

186.  Let  the  polygonal  line  ABC  .  .  .  YZ,  of  an  infinite 
number  of  sides,  be  taken  to  represent 
,  the  curved  portion  of  a  cord  embracing 
1  any  arc  of  a  cylindrical  surface  (whe- 
ther circular  or  not),  in  a  plane  per- 
pendicular to  the  axis  of  the  cylinder ; 
also  let  Aa,  B5,  Cc,-  &c.,  be  normals 
or  perpendiculars  to  the  curve,  inclined 
to  one  another  at  equal  angles,  each 
represented  by  Ad.  Imagine  the  surface  of  the  cylinder  to 
be  removed  between  each  two  of  the  points  A,  B,  &c.,  in 
succession,  so  that  the  cord  may  be  supported  by  a  small 
portion  only  of  the  surface  remaining  at  each  of  those 
points,  whilst  in  the  intermediate  space  it  assumes  the  direc- 
tion of  a  straight  line  joining  them,  and  does  not  touch  the 
surface  of  the^  cylinder.  Let  P,  represent  the  tension  upon 
the  cord  before  "it  has  passed  over  the  point  A  ;  T1  the  ten- 
sion upon  it  after  it  has  passed  over  that  point,  or  before  it 
passes  over  the  point  B  ;  T2  the  tension  upon  it  after  it  has 
passed  over  the  point  B,  or  before  it  passes  over  C  ;  T8  that 
after  it  has  passed  over  C ;  and  let  P2  represent  the  tension 
upon  the  cord  after  it  has  passed  over  the  nth  or  last 
point  Z. 

Now,  any  point  B  of  the  cord  is  held  at  rest  by  the  ten- 
sions Tt  and  T2  upon  it  at  that  point,  in  the  directions  BC 
and  BA,  and  by  the  resistance  K  of  the  surface  of  the  Cylin- 
der there ;  and,  if  we  conceive  the  cord  to  be  there  in  the 
state  bordering  upon  motion,  then  (Art,  138.)  the  direction 
of  this  resistance  K  is  inclined  to  the  perpendicular  5B  to 
the  surface  of  the  cylinder  at  an  angle  RB&  equal  to  the 
limiting  angle  of  resistance  <p. 


208  THE  FKICTION   OF   CORDS. 

Now  T,,  T2,  and  E  are  pressures  in  equilibrium ;   there- 
fore (Art.  14.) 

T,_jaiiL  TJBE 
Ta~sin.  T,BE ' 

but  T1BE=AB5-EBJ=i(*-A^B)-EB5)=  ~-~  -9, 

2          A 
if         Ad 


sm.^- 


"T 


COS.    \-7T 


0     .       Ad     . 

2  sin.  —  sin.  9 


Ad  .       Ad    . 

cos.  —  cos.  9  —  sin.  —  sin.  9 


or  dividing  numerator  and  denominator  of  the  fraction  in  the 

Ad  i 

second  member  by  cos.  -~-  cos.  9, 


T.-T, 


Ad 

2  tan.  —  tan.  9 


"a          1— tan.-—  tan.  9 


Suppose  now  the  angles  A#5,  BZ>C,  &c.,  each  of  which 
equals  Ad,  to  be  exceedingly  small,  and  therefore  the  points 
A,  B,  C,  &c.,  to  be  exceedingly  near  to  one  another,  and 
exceedingly  numerous.  By  this  supposition  we  shall  mani- 
festly approach  exceedingly  near  to  the  actual  case  of  an  in- 
finite number  of  such  points  and  a  continuous  surface ;  and 


THE   FRICTION   OF   COEDS.  209 

if  we  suppose  Ad  infinitely  small,  our  supposition  will  coincide 
with  that  case.     Now,  on  the  supposition  that  Ad  is  exceed- 

Ad 

ingly  small,   tan.  -^  .  tan.  9  is  exceedingly  small,  and  may 

be  neglected  as  compared  with  unity  ;  it  may  therefore  be 
neglected  in  the  denominator  of  the  above  fraction.     More- 

over Ad  being  exceedingly  small,  tan.  -^-  =  — 


,-S 

•*•      Ta      =  tan.  9  .  Ad*  ;    .-.  T1=T3  (1  +  tan.  9  .  Ad). 

Now  the  number  of  the  points  A,  B,  C,  &c.  being  repre- 
sented by  7i,  and  the  whole  angle  AdZ  between  the  extreme* 
normals  at  A  and  Z  by  d,  it  follows  (Euclid,  i.  32.)  that 

6—n.  Ad;  therefore  Ad  =-; 
7i 


n 


Similarly,  P^T,  (1  +-tan.  9) 

n 


T,=T,  (l+tan.  ?), 


^P,  (1+tan.  9). 


Multiplying  these  equations  together,  and  striking  out  fac- 
tors common  to  both  sides  of  their  product,  we  haye 


*  If  we  consider  the  tension  T  as  a  function  of  6,  of  which  any  consecutive 
values  are  represented  by  Tj  and  T2,  and  their  difference  or  the  increment  of 

L *  rn  -«  AT* 

T  by  AT,  then  — ^—  =  tan.  0.  A0 ;  therefore  -  .  — ^    =  —  tan.  0 ;  therefore, 

passing  to  the  limit  -  —  =  —  tan.  0,  and  integrating  between  the  limits  0 
and  6,  observing  that  at  the  latter  Limit  T=P2,  and  that  at  the  former  it  equals 
PI,  we  have  log.  ( —  1  =  -  6  tan.  0;  therefore  P1=P^6  tan"  *. 

\TI/ 

H 


210  THE   FKICTION   OF   CORDS. 

(  — 

or  P1=Pa  \  1 

\ 


n—  In—  26'  \ 

»-a  --  r-tan.V  +  &c.  f; 

(  !-; 

orP1=Pa|l+4  tan.  9  +  —  ^a  tan.  a<p  + 


Now  this  relation  of  Px  and  P3  obtains  however  small  Ad 
be  taken,  or  however  great  n  be  taken.  Let  n  be  taken 
infinitely  great,  so  that  the  points  A,  B,  C,  &c.  may  be 
infinitely  numerous  and  infinitely  near  to  each  other.  The 
supposed  case  thus  passes  into  the  actual  case  of  a  con- 

•i    2   Q 
tinuous  surface,  the  fractions  -,  -,  -,  &c.  vanish,  and  the 

above  equation  becomes 

6  tan.  9     da  tan.  "9     &*  tan.  '<p  ) 

But  the  quantity  within  the  brackets  is  the  well  known  ex- 
pansion (by  the  exponential  theorem)  of  the  function  efltaa-d, 

*•* (205). 


Since  the  length  of  cord  S,,  which  passes  over  the  point 
A,  is  the  same  with  that  S2  -which  passes  over  the  point  Z, 
it  follows  that  the  modulus  (Art.*  152.)  of  such  a  cylindrical 
surface  considered  as  a  machine,  and  supposed  to  \>t  fixed 
and  to  have  a  rope  pulled  and  made  to  slip  over  it,  is 

U^-Q^tan.0  ____  (206). 

It  is  remarkable  that  these  expressions  are  wholly  inde- 
pendent of  the  form  and  dimensions  of  the  surface  sustain- 
ing the  tension  of  the  rope,  and  that  they  depend  exclu- 
sively upon  the  inclination  6  or  Ae£Z  of  the  normals  to  the 
points  A  and  Z,  where  the  cord  leaves  the  surface,  and  upon 
the  co-efficient  of  friction  (tan.  9),  of  the  material  of  which 
the  rope  is  composed  and  the  material  of  which  the  surface 
is  composed.  It  matters  not,  for  instance,  so  far  as  ihefric- 


THE   FKICTION   OF   CORDS.  211 

t-ion  of  the  rope  or  band  is  concerned,  whether  it  passes 
over  a  large  pulley  or  drum,  or  a  small  one,  provided  the 
angle  subtended  by  the  arc  which  it  embraces  is  the  same, 
and  the  materials  of  the  pulley  and  rope  the  same. 

In  the  case  in  which  a  cord  is  made  to  pass  m  times  round 
such  a  surface,  G=9m7r ; 

•  T>  — p  c2m  TT  tan.  & 
»*•*•)  —  x  <r  r» 

And  this  is  true  whatever  be  the  form  of  the  surface,  so 
that  the  pressure  necessary  to  cause  a  cord  to  slip  when 
wound  completely  round  such  a  cylindrical  surface  a  given 
number  of  times  is  the  same  (and  is  always  represented  by 
this  quantity),  whatever  may  be  the  form  or  dimension  of 
the  surface,  provided  that  its  material  be  the  same.  It 
matters  not  whether  it  be  square,  or  circular,  or  elliptical. 


1ST.  If  P/,  P/',  P/",  &c.  represent  the  pressures  which 
must  be  applied  to  one  extremity  of  a  rope  to  cause  it  to 
slip  when  wound  once,  twice,  three  times,  &c.  round  any 
such  surface,  the  same  tension  P2  being  in  each  case  sup- 
posed to  be  applied  to  the  other  extremity  of  it,  we  have 


So  that  the  pressures  P/,  P/',  P/",  &c.  are  in  a  geome- 
trical progression,  whose  common  ratio  is  e^tan.^  which 
ratio  is  always  greater  than  unity.  Thus  it  appears  by  the 
experiments  of  M.  Morin  (p.  135.),  that  the  co-efficient  of 
friction  between  hempen  rope  and  oak  free  from  unguent  is 
•33,  when  the  rope  is  wetted.  In  this  case  tan.  9=  -33  and 
27rtan.  <p=2  x3-U159x  -33=2-07345.  The  common  ratio 
of  the  progression  is  therefore  in  this  case  e2'07345,  or  it  is  the 
number  whose  hyperbolic  logarithm  is  2-07345.  This  num- 
ber is  7*95  ;  so  that  each  additional  coil  increases  the  fric- 
tion nearly  eight  times.  Had  the  rope  been  dry,  this 
proportion  would  have  been  much  greater.  If  an  additional 
half  coil  had  been  supposed  continually  to  be  put  upon  the 
rope  instead  of  a  whole  coil,  the  friction  would  have  been 
found  in  the  same  way  to  increase  in  geometrical  progres- 
sion, but  the  common  ratio  would  in  this  case  have 
been  e71"**11-?  instead  of  e27rtan.^  jn  ^ne  above  example  the 
value  of  this  ratio  would  for  each  half  coil  have  been 
2-82. 

The  enormous  increase  of  friction  which   results  from 


212  THE  FRICTION   OF  COEDS. 

each  additional  turn  of  the  cord  upon  a  capstan  or  drum 
may  from  these  results  be  understood. 


188.  We  may,  from  what  has  been  stated  above,  readily 
explain  the  reason  why  a  knot  connecting  the  two  extremi- 
ties of  a  cord  effectually  resists   the   action  of  any  force 
tending  to  separate  them.     If  a  wetted  cord  be  wound  round 
Fig.  i.  mg.  2.  Fig.  s.         a  cylinder  of  oak  as 

in  fig.  1.,  and  its  ex- 
tremities be  acted 
upon  by  two  forces  P 
and  R,  it  has  been 
shown  that  P  will  not 
overcome  R,  unless  it 
be  equal  to  some- 
where about  eight  times  that  force.  Now  if  the  string  to 
which  R  is  attached  be  brought  underneath  the  other  string 
so  as  to  be  pressed  by  it  against  the  surface  of  the  cylinder, 
as  at  m,fig.  2.;  then,  provided  the  friction  produced  by 
this  pressure  be  not  less  than  one  eighth  of  P,  the  string  will 
not  move  even  although  the  force  R  cease  to  act.  And  if 
both  extremities  of  the  string  be  thus  made  to  pass  between 
the  coil  and  the  cylinder,  as  in  fig.  3.,  a  still  less  pressure 
upon  each  will  be  requisite.  Now,  by  diminishing  the 
radius  of  the  cylinder,  this  pressure  can  be  increased  to  any 
extent,  since,  by  a  known  property  of  funicular  curves,  it 
varies  inversely  as  the  radius.*  We  may,  therefore,  so  far 
diminish  the  radius  of  a  cylinder,  as  that  no  force,  however 
great,  shall  be  able  to  pull  away  a  rope  coiled  upon  it,  as 
represented  in  fig.  3.,  even  although  one  extremity  were 
loose,  and  acted  upon  by  no  force. 

Fig- 4t  Let  us  suppose  the  rope  to  be 

doubled  as  in  fig.  4.,  and  coiled 
as  before.  Then  it  is  apparent, 
from  what  has  been  said,  that 
the  cylinder  may  be  made  so 
small,  that  no  forces  P  and  P' 
applied  to  the  extremities  of 
either  of  the  double  cords  will 
be  sufficient  to  pull  them  from 
it,  in  whatever  directions  these  are  applied. 

*  This  property  will  be  proved  in  that  portion  of  the  work  which  treats  of 
the  THEORY  OF  CONSTRUCTION. 


THE   FRICTION    BREAK.  213 

Now  let  the  cylinder  be  removed.  The  cord  then  being 
drawn  tight,  instead  of  being  coiled  round  the  cylinder,  wifl 
be  coiled  round  portions  of  itself,  at  the  points  m  and  n ; 
and  instead  of  being  pressed  at  those  points  upon  the  cylin- 
der, by  a  force  acting  on  one  portion  of  its  circumference,  it 
will  be  pressed  by  a  greater  force  acting  all  round  its  cir- 
cumference. All  that  has  been  proved  before,  with  regard 
to  the  impossibility  of  pulling  either  of  the  cords  away  from 
the  coil  when  the  cylinder  is  inserted,  will  therefore  now 
obtain  in  a  greater  degree  ;  whence  it  follows  that  no  forces 
P  and  P'  acting  to  pull  the  extremities  of  the  cords  asunder, 
may  be  sufficient  to  separate  the  knot. 


THE  FRICTION  BREAK. 

189.  There  are  certain  machines  whose  motion  tends,  at 
certain  stages,  to  a  destructive  acceleration ;  as,  for  instance, 
a  crane,  which,  having  raised  a  heavy  weight  in  one  position 
of  its  beam,  allows  it  to  descend  by  the  action  of  gravity  in 
another ;  or  a  railway  train,  which,  on  a  certain  portion  of 
its  line  of  transit,  descends  a  gradient,  having  an  inclination 
greater  than  the  limiting  angle  of  resistance.  In  each  of 
these  cases,  the  work  done  by  gravity  on  the  descending 
weight  exceeds  the  work  expended  on  the  ordinary  resist- 
ance due  to  the  friction  of  the  machine ;  and  if  some  other 
resistance  were  not,  under  these  circumstances,  opposed  to 
its  motion,  this  excess  (of  the  work  done  by  gravity  upon  it 
over  that  expended  upon  the  friction  of  its  rubbing  surfaces) 
would  be  accumulated  in  it  (Art.  130.)  under  the  form  of 
vis  viva,  and  be  accompanied  by  a  rapid  acceleration  and  a 
destructive  velocity  of  its  moving  parts.  The  extraordinary 
resistance  required  to  take  up  its  excess  of  work,  and  to 
prevent  this  accumulation,  is  sometimes  supplied  in  the 
crane  by  the  work  of  the  laborer,  who,  to  let  the  weight 
down  gradually,  exerts  upon  the  revolving  crank  a  pressure 
in  a  direction  opposite  to  that  which  he  used  in  raising  it. 
It  is  more  commonly  supplied  in  the  crane,  and  always  in 
the  railway  train,  without  any  work  at  all  of  the  laborer,  by 
a  simple  pressure  of  his  hand  or  foot  on  the  lever  of  the  fric- 
tion break,  which  useful  instrument  is  represented  in  the 
figure  under  the  form  in  which  it  is  com 


-l  THE   FRICTION   BEEAK. 

monly  applied  to  the  crane, — a  form  of  it  which  may  serve 
to  illustrate  the  principle  of  its  application  nnder  every 

other.  BC  represents  a  wheel 
fixed  commonly  upon  that 
axis  of  the  machine  to  which 
the  crank  is  attached,  and 
which  axis  is  carried  round 
by  it  with  greater  velocity 
than  any  other.  The  peri- 
phery of  this  wheel,  which  is 
usually  of  cast  iron,  is  em- 
braced by  a  strong  band*  ABCE  of  wrought  iron,  fixed 
firmly  by  its  extremity  A  to  the  frame  of  the  machine,  and 
by  its  extremity  E  to  the  short  arm  AE  of  a  bent  lever  PAE, 
which  turns  upon  a  fixed  axis  or  fulcrum,  at  A,  and  whose 
arm  PA,  being  prolonged,  carries  a  counterpoise  D  just 
sufficient  to  overbalance  the  weight  of  the  arm  AP,  and  to 
relieve  the  point  E  of  all  tension,  and  loosen  the  strap  from 
the  periphery  of  the  wheel,  when  no  force  P  is  applied  to  the 
extremity  of  the  arm  AP,  or  when  the  break  is  out  of 
action. 

It  is  evident  that  a  pressure  P  applied  to  the  extremity  of 
the  lever  will  produce  a  pressure  upon  the  point  E,  and  a 
tension  upon  the  band  in  the  direction  ABCE,  and  that 
being  fixed  at  its  extremity  A,  the  band  will  thus  be  tight- 
ened upon  the  wheel,  producing  by  its  friction  a  certain 
resistance  upon  the  circumference  of  the  wheel. 

Moreover,  it  is  evident  that  this  resistance  of  friction  upon 
the  circumference  of  the  wheel  is  precisely  equal  to  the 
tension  upon  the  extremity  A  of  the  band,  being,  indeed, 
wholly  borne  by  that  tension ;  •  and  that  it  is  the  same 
whether  the  wheel  move,  as  in  this  case  it  does,  under  the 
band  at  rest,  or  whether  the  band  move  (under  the  same 
tensions  upon  its  extremities,  but  in  the  opposite  direction) 
over  the  wheel  at  rest.  Let  R  and  Q  represent  the  tensions 
upon  the  extremities  A  and  E  of  the  band ;  then  if  we  sup- 
pose the  wheel  to  be  at  rest,  and  the  band  to  be  drawn  over 
it  in  the  direction  ECB  by  the  tension  R,  and  &  to  represent 
the  angle  subtended  at  the  centre  of  the  wheel  by  that  part 
of  its  circumference  which  the  band  embraces,  we  have 
(equation  205) 

*  Blocks  of  wood  arc  interposed  between  the  band,  the  periphery  of  the 
break  wheel.  This  case  will  be  discussed  in  the  Appendix. 


THE  BAND.  215 


Let  &J  represent  the  length  of  the  arm  AP,  and  #a  the 
length  of  the  perpendicular  let  fall  from  A  upon  the  direc- 
tion of  a  tangent  to  that  point  in  the  circumference  of  the 
wheel  where  the  end  EC  of  the  band  leaves  it. 

Then,  neglecting  the  friction  of  the  axis  A,  we  have 
(Art.  5.) 

P  .  a=     .  0 


(207). 


If  Pj  represent  any  pressure  applied  to  the  circumference^  of 
the  break  wheel,  and  P2  a  pressure  applied  to  the  working 
point  of  the  machine,  whatever  it  may  be,  to  which  the 
break  is  applied,  and  if  Yl=aP^-\-b  (Art.  152.)  represent  the 
relation  between  Pt  and  P2  in  the  inferior  state  bordering 
upon  motion  by  the  preponderance  of  P2  ;  then,  when  P2  is 
taken  in  this  expression  to  represent  the  pressure  W,  whose 
action  upon  the  working  point  of  the  machine  the  break  is 
intended  to  control,  'Pl  will  represent  that  value  K  of  the 
friction  upon  the  break  which  must  be  produced  by  the 
intervention  of  the  lever  to  control  the  action  of  the  pressure 
W  upon  the  machine;  so  that  taking  E  to  represent  the 
same  quantity  as  in  equation  (207),  we  have 


Eliminating  E-  between  this  equation  and  equation  (207), 
and  solving  in  respect  to  P, 


....  (208). 
a.% 


THE  BAND. 


1 90.  When  the  circular  motion  of  any  shaft  in  a  machine, 
and  the  pressure  which  accompanies  that  motion,  consti- 
tuting together  with  it  the  work  of  the  shaft,  are  to  be  com- 
municated to  any  other  distant  shaft,  this  communication  is 


216  THE   BAND. 

usually  established  by  means  of  a  band  of 
leather,  which  passes  round  drums  fixed  upon 
the  two  shafts,  and  has  its  extremities  drawn 
together  with  a  certain  pressure  and  united, 
so  as  to  produce  a  tension,  which  should  be 
just  that  necessary  to  prevent  the  band  from 
slipping  upon  the  drums,  subject  to  the  pres- 
sure under  which  the  work  is  transferred. 
The  facility  with  which  this  communication 
of  rotatory  motion  may  be  established  or 
broken  at  any  distance  and  under  almost 
every  variety  of  circumstance,  has  brought 
the  band  so  extensively  into  use  in  machinery, 
that  it  may  be  considered  as  a  principal  chan- 
nel through  which  work  is  made  to  flow  in  its  distribution 
to  the  successive  stages  of  every  process  of  mechanism, 
carried  on  in  the  same  workshop  or  manufactory. 


191.  The  sum  of  the  tensions  upon  the  two  parts  of  a  hand 
is  the  same,  whatever  he  the  pressure  under  which  the  hand 
is  d/riven,  or  the  resistance  overcome,  the  tension  of  the 
driving  part  of  the  hand  being  always  increased  hy  just  so 
much  as  that  of  the  driven  part  is  diminished. 

This  principle  was  first  given  by  M.  Poncelet ;  it  has  since 
been  amply  confirmed  by  the  experiments  of  M.  Morin.*  It 
may  be  proved  as  followsf  : — In  the  very  commencement  of 
the  motion  of  that  drum  to  which  the  driving  pressure  is 
applied,  no  motion  is  communicated  by  it  to  the  other  drum. 
Before  any  such  motion  can  be  communicated  to  the  latter, 
a  difference  must  be  produced  between  the  tensions  of  the 
two  parts  of  the  band  sufficient  to  overcome  the  resistance, 
whatever  it  may  be,  which  is  opposed  to  the  revolution  of 
the  driven  drum.  Now,  an  increase  of  the  tension  on  the 
driving  side  of  the  band  must  be  followed  by  an  elongation 
of  that  side  of  the  band  (since  the  band  is  elastic),  and  by 
the  revolution  of  the  circumference  of  the  driving  drum 
through  a  space  precisely  equal  to  this  elongation.  Sup- 
posing, then,  the  other,  or  driven  side  of  the  band,  to 
remain  extended,  as  before,  in  a  straight  line  between  its  two 
points  of  contact  with  the  drums,  this  portion  of  the  band 

*  Nouvelles  Experiences  sur  le  Froltement,  &c.     Metz. 

f  No  demonstration  appears  to  have  been  given  of  it  by  M.  Poncelet, 


THE   BAND. 


217 


must  evidently  have  contracted  by  precisely  the  length 
through  which  the  circumference  of  the  driving  drum  has 
revolved,  or  the  driving  side  of  the  band  elongated.  Thus, 
the  elongation  of  the  driving  side  of  the  band  is  precisely 
equal  to  the  contraction  of  the  driven  side.  Now,  the  band 
being  supposed  perfectly  elastic,  the  increase  or  dimi- 
nution of  its  tension  is  exactly  proportional  to  the  increase 
or  diminution  of  its  length.  The  increase  of  tension  on  the 
one  side,  produced  by  a  given  elongation,  is  therefore  pre- 
cisely equal  to  the  diminution  of  tension  produced  by  a  con- 
traction equal  to  that  elongation  on  the  other  side.  Thus, 
if  T  represent  the  tension  upon  each  side  of  the  band  before 
the  driving  pressure,  whatever  it  may  be,  was  applied, 
and  if  T,  and  T2  represent  the  tensions  upon  the  driving 
and  the  driven  sides  of  the  band  after  that  pressure  is 
applied;  then,  since  Tx—  T  represents  the  increase  of  tension 
on  the  one  side,  and  T—  T2  the  diminution  of  tension  on  the 
other,  ^-T^T-T,; 


(209). 


It  is  a  great  principle  of  the  economy  of  power  in  the  use 
of  the  band  to  adjust  this  initial  tension  T,  so  that  it  may 
just  be  sufficient  to  prevent  the  band  from  slipping  upon 
the  drum  under  any  pressure  which  it  is  required  to  transmit. 
The  means  of  making  this  adjustment  will  be  explained 
hereafter. 

THE  MODULUS  OF  THE  BAND. 

192.  For  simplifying  the  consideration  of  this  important 
element  in  machinery,  we  shall  first  consider  a  particular 
case  of  its  application.  Let  the  two  drums^  whose  axis  are 
G!  and  C2,  be  supposed  equal  to  one  another,  so  that  the  two 
parts  of  the  band  which  pass  round  them  may  be  parallel. 

Let,  moreover,  the  centres  of  the 
two  drums  be  in  the  same  verti- 
cal straight  line,  so  that  the  two 
parts  of  the  band  may  be  verti- 
cal. 

Let  Pj  and  P2  be  pressures  ap- 
plied, in  vertical  directions,  to 
turn  the  drums,  and  at  perpen- 
dicular distances  from  their  cen- 
tres, represented  by  GXP>  and 
C2P2;  of  which  pressures  P2  is 
the  working  or  driven  pressure. 


Fig  2 


218  THE    BA^D. 

or  that  which  is  upon  the  point  of  yielding  by  the  prepon- 
derance of  the  other  P,.  In  fig.  1.  P2  is  seen  applied  on 
the  same  side  of  the  centre  of  the  drums  as  P15  and  in  jig. 
2.  on  the  opposite  side.  Let  Tx  and  T2  represent  the  tensions 
upon  the  two  parts  of  the  band,  Tx  being  that  on  the  driving, 
and  T2  that  on  the  driven  side. 

^=0^,  «2=C2P2, 

r=  radius  of  each  drum, 

W=  weight  of  each  drum, 

p=  radius  of  axis  of  each  drum, 

R!  and  R2=  resistances  of  axes  of  drums, 

<p—  limiting  angle  of  resistance. 

Now,  the  parallel  pressures  P,,  W,  T15  T2,  Rn  applied  to  the 
lower  drum,  are  in  equilibrium  ;  therefore  (Art.  16.), 


or  substituting  for  T^+T,  its  value  2T  (equation  209), 
E1=±(2T—  P1—  W)  .....  (210). 

The  sign  ±  being  taken  according  as  2T  is  greater  or  less 
than  Pj+Wj,  or  according  as  the  axis  of  the  lower  drum 
presses  upon  the  upper  surface  of  its  bearings,  as  shown  in 
fig.  1.,  or  upon  the  lower  surface,  as  shown  v&fig.  2.  In  like 
manner,  the  pressures  P2,  W,  T1?  T2  R2,  applied  to  the  upper 
drum,  being  in  equilibrium, 


or  (equation  209)  R2=2TTPa+W  ....  (211), 

where  the  sign  ^p  is  to  be  taken  according  as  P2  is  applied 
on  the  same  side  of  the  axis  as  P,,  or  on  the  opposite  side. 

Since,  moreover,  Rj  and  R2  act,  in  the  state  bordering 
upon  motion,  at  perpendicular  distances  from  the  centre  of 
the  axis,  which  are  each  represented  by  p  sin.  <p  (Art.  153.), 
we  have,  by  the  principle  of  the  equality  of  moments, 

PA+T,r=Tlr+:Rlpsm.9 


observing  that  the  resultant  of  all  the  pressures  applied  to 
each  drum  (excepting  only  the  resistance  of  its  axis)  must  be 
such  as  would  alone  communicate  motion  to  it  in  the  direc- 
tion in  which  it  actually  moves,  and  therefore  that  the  re 
sistance  of  the  axis,  which  is  opposite  to  this  resultant,  must 
tend  to  communicate  motion  to  the  drum  in  a  direction  oppo- 
site to  that  in  which  it  actually  moves. 


THE   BAND.  .219 

Subtracting  the  above  equations,  and  transposing, 
P1«,-PA=(RI+R2)  p  sin.  9. 

Substituting  the  values  of  Rx  and  Ra  from  equations  (210) 
and  (211),  we  obtain,  in  the  case  in  which  the  negative  sign 
of  Rj  is  to  be  taken,  or  in  which  2T  is  less  than  Pa  +  W,  the 
axis  Cx  resting  upon  the  lower  surface  of  its  collar  as  shown 
in  fig.  2., 

sin.  9  ; 


and  in  the  case  in  which  the  positive  sign  of  3^  is  to  be 
taken,  2T  being  greater  than  PX+W,  and  the  axis  Ct  press- 
ing against  the  upper  surface  of  its  collar,  as  shown  iRJtg.  1., 


sn.  9. 

Transposing  and  reducing,  we  obtain  for  the  relation  be- 
tween the  driving  and  driven  pressures  in  these  two  cases 
respectively, 

P     P  /«sSL«2i£\    2WPBin.,>  .       .  (213), 

\a1—  p  sin.  9;     at—  psm.  9 


p      p  .  (214) 

\a1  +  psm.  97 


and  therefore  (by  equation  121),  for  the  moduli  in  the  two 
casesa 


!—  p  sn.  9 


sin.  0 


In  all  which  equations  the  sign  =F  is  to  be  taken  according 
as  P2  is  applied  on  the  same  side  of  the  line  OjC,,  joining  the 
axis  as  P15  or  on  the  opposite  side. 


193.  To  determine  the  initial  tension*^  upon  the  land,  so  that 
it  may  not  slip  upon  the  surface  of  the  drum  when  sub- 
jected  to  the  given  resistance  opposed  to  its  motion  by  the 
work. 


220 


THE   BAND. 


Suppose  the  maximum  resistance  which  may,  during  the 
action  of  the  machine,  be  opposed  to  the  motion 
of  the  drum  to  be  represented  by  a  pressure  P 
applied  at  a  given  distance  a  from  its  centre  C2. 
Suppose,  moreover,  that  the  band  has  received 
such  an  initial  tension  T  as  shall  just  cause  it  to 
be  on  the  point  of  slipping  when  the  motion  of 
the  drum  is  subjected  to  this  maximum  resist- 
ance ;  and  let  ^  and  tz  be  the  tensions  upon 
the  two  parts  of  the  band  when  it  is  thus 
Just  in  the  act  of  slipping  and  of  overcoming  the  resistance 
r.  Now,  the  two  parts  of  the  band  being  parallel,  it  em 
braces  one  half  of  the  circumference  of  each  drum  ;  the  rela- 
tion between  tl  and  £2  is  therefore  expressed  (equation  205) 
by  the  equation 

TT  tan.  ^ 

t  =  t ae*-tan. 0  'whence  we  obtain  -1 — J-=--  ~     .  But  £.  +  £,= 

1  *  '•  -r        I      •/  ,— -    to^»      A.  *  * 


2T  (equation  209), 


(TT  tan.  0 
e—^ 
it  tan.  ^ 
e    +    I 


Also,  the  relation  between  the  resistance  P,  opposed  to  the 
motion  of  the  upper  drum,  and  the  tensions  ^  and  £2  upon 
the  two  parts  of  the  band,  when  this  resistance  is  on  the 
point  of  being  overcome,  is  expressed  (equation  212)  by  the 
equation 

or  substituting  the  value  of  Ra  (equation  211),  and  transpos- 
ing 

P«+(2T=f  P-f  W)p  sin.  9=)^— t^r  ; 

whence,  substituting  the  value  of  ^—4,  determined  above, 
and  transposing,  we  have 

{/e"-'\\  } 

P(aqFpsin.  9)H-"Wpsin.  9=2T^  |  -^^-^  \r— psin.9  j. ; 

e    +    I 


THE   BAND. 


T_ -  J  P(a=Fp  sin.  0)  +  Wpsin.0  ) 

*   i       /    TT  tan.  0  \ 
I  8     —     1\ 

I  -¥tS^  l^-psm.  </> 


221 


(217). 


194.  The  modulus  of  the  band  under  its  most  general  form, 

The  accompanying  figure  represents  an.  elastic  band  pass- 
ing round  drums  of  unequal  radii,  the 
line  joining  whose  centres  0,  and  C2 
is  inclined  at  any  angle  to  the  vertical, 
and  which  are  acted  upon  by  any 
given  pressures  P,  and  P2,  P1  being 
supposed  to  be  upon  the  point  of  giv- 
ing motion  to  the  system. 

Let  Tx  and  Ta  represent  the  tensions 
upon  the  two  parts  of  the  band,  Tx  be- 
ing that  on  the  driving  side. 
#15  #2  perpendiculars  upon  the  directions  of  Pa  and  Pa  re- 
spectively. 

01?  02  the  inclinations  of  the  directions  of  P,  and  P2  to  the 
line  0,0,. 

TV  r2  the  radii  of  the  drums. 
W  j,  W2  the  weights  of  the  drums. 

«  the  inclination  of  the  line  Cfl9  to  the  vertical,  and2ajthe 
inclination  of  the  two  parts  of  the  band  to  one  another, 
p!  p2  the  radii  of  the  axes  of  the  drums. 
0  the  limiting  angle  of  resistance  between  the  axis  of  the 
drum  and  its  collar. 

K15  R2  the  resistances  of  the  collars  in  which  the  axes  of 
the  drums  turn  in  the  state  bordering  upon  motion,  or  the 
resultants  of  the  pressures  upon  these  axes.  The  perpendi- 
cular distances  at  which  these  resistances  act  from  the  cen- 
tres of  the  axes  are  (Art.  153.)  p:  sin.  0  and  p2  sin.  0.  Since 
the  pressures  acting  upon  the  lower  drum  are  T15  T2,  P1?  Wiy 
and  Kj,  and  that  these  pressures  are  in  equilibrium,  Wl  act- 
ing through  the  centre  of  the  axis,  and  T,  and  Et  acting  to 
turn  the  drum  in  one  direction  about  the  axis,  and  Px  and  T, 
to  turn  it  in  the  opposite  direction  ;  we  have,  by  the  princi- 
ple of  the  equality  of  moments  (Art.  153.), 

PA+T^^T^  +  K^sin.  9. 
And  since  T1?  T2,  P2,  W2,  Ea  are  similarly  in  equilibrium 


222  THE   BAND. 

on  the  upper  drum,  W2  acting  through  the  centre,  and  P2, 
R2,  T2  acting  to  turn  it  in  one  direction,  whilst  Tl  acts  to 
turn  it  in  the  opposite  direction, 

sin.  9= 


/.PA-^-T^K^sin.?      ) 

pA_(Tl-T2K=-K2p2  sin.  9  T 
Let  Tj-T^B*,  and  T.+T^^T, 


^^R^  sin.  9      )  ,01  c^ 

-2«rf=-  R2p,  sin.  9  (' 

To  determine  the  values  of  1^  and  R2  let  the  pressures 
applied  to  each  drum  be  resolved  (Art.  11.)  in  directions 
parallel  and  perpendicular  to  the  line  CjC,  ;  tnose  applied  to 
the  lower  drum  which,  being  thus  resolved,  are  parallel  to 
00  are 


!  COS.  «15    +Ta  COS.  ai5   —  P!  COS.  ^,   —  Wl  COS.  I, 

those  pressures  being  taken  positively  which  tend  to  move 
the  axis  of  the  drum  from  Gl  towards  02,  and  those  nega- 
tively whose  tendency  is  in  the  opposite  direction. 

In  like  manner  the  pressures  resolved  perpendicular  to 
0,0,  are 

—  Tx  sin.  al?  -j-T2  sin.  al?  -f  P4  sin.  ^,  —  "W1  sin.  », 

those  pressures  being  taken  negatively  whose  tendency 
when  thus  resolved  perpendicular  to  C^  is  to  bring  that 
line  nearer  to  a  vertical  direction,  and  those  positively  whose 
tendency  is  in  the  opposite  direction. 

Observing  that  K1  is  the  resultant  of  all  these  pressures, 
we  have  (Art.  11.) 

os-  ai-pi  cos-  *i-Wi  cos-  '}*  + 


|PX  sin.  ^-(T.-T^sin.  a.-W,  sin.  i}2. 

Proceeding  similarly  in  respect  to  the  pressures  applied  to 
the  upper  drum,  we  shall  obtain 

Ea3=  {(T.+T,)  cos.  «,—  P,  cos.  ^a+W2  cos.  i}f+ 

|P3  sin.  ^+(T1—T3)  sin.  a,—  W2  sin.  »J3  : 
or  substituting  2T  for  T^T,,  and  2#  for  T,—  T2 

K^  |2T  cos.  «,—  P,  cos.  *!—  W,  cos.  »}2 
{P,  sin.  ^-2^  sin.  ^-W,  sin.  •{  3 

Ra9=  J2T  cos.  ^-P,  cos.  ^a+  W2  cos.  1}  '+  ' 
}Pa  sin.  d2  +  2£  sin.  a,—  W2  sin.  1}  2 


THE   BAND. 


By  eliminating  R15  R2,  and  t  between  the  four  equations 
(218)  and  (219),  a  relation  is  determined  between  the  three 
quantities  P15  P2,  T.  To  simplify  this  elimination  let  us  sup- 
pose that  the  preceding  hypothesis  in  respect  to  the  direc- 
tions in  which  the  pressures  are  to  be  taken  positively  and 
negatively  is  so  made,  that  the  expressions  enclosed  within 
the  brackets  in  the  above  equations  (219)  and  squared  may, 
each  of  them,  represent  a  positive  quantity.  Let  us,  more- 
over, suppose  t\iQ  first  of  the  two  quantities  squared  in  each 
equation  to  be  considerably  greater  than  the  second,  or  the 
pressure  upon  the  axis  of  each  drum  in  the  direction  of  the 
line  Ca  C2  joining  their  centres,  greatly  to  exceed  the  pres- 
sure upon  it  in  a  direction  perpendicular  to  that  line  ;  an 
hypothesis  which  will  in  every  practical  case  be  realised. 
These  suppositions  being  made,  we  obtain,  with  a  sufficient 
degree  of  approximation,  by  Poncelet's  Theorem*, 

Ri:=aj2T  cos.  ax—  Pt  cos.  0,—W,  cos.  i{  + 
/3  ?  sin.  6—2t  sin.  a—  W  sin.  « 


R2=a{2T  cos.  al—  P2  cos.  03+W3  cos.  1} 
/3  JP2  sin.  02+22  sin.  ax—  W2  sin.  1}  . 


Substituting  these  values  of  Rj  and  Ra  in  equation  (218), 
and  reducing,  we  have 

PA—  2£(r\—  /3pl  sin.  at  sin.  9)= 
Pl  |2«T  cos.  ai-PA-WlTll  sin.  9 

P2a2-2^2-/3p2  sin.  ttl  sin.  9)= 
-p9{2a  T  cos.  a.-PA+'W^I  sin.  9 

where  ft=(a  cos.  0X—  ]8  sin.  0X), 
ft=(.  co,  ^_sin  *,), 
y1=(a  cos.  i  +  P  Sin.  <), 
y3=(a  cos.  i—  £  sin.  i). 

Eliminating  #  between  these  equations,  and  neglecting 
terms  above  the  first  dimension  in  p1  sin.  9  and  p2  sin.  9, 

(  H-P^Xn—  £pa  sin.  ax  sin.  9)  |  _ 
(  —  P^r,—  £pa  sin.  a,  sin.  9)  j  " 


+Plr,(2aT  cos.  ^-PA-W^)  )    . 

T,)  [  Sm' 


+P2rX2aT  cos.  a.- 
,  being  for  the  most  part  exceedingly  small,  the  terms 

*  See  Appendix. 


£24 


THE   BAND. 


/3pj  sin.  ctj  sin.  9  and  /3pa  sin.  az  sin.  9  may  be  neglected ;  we 
shall  then  obtain  by  transposition  and  reduction 


+2*T(p1riH-ptr1)cos.  at 


-sin.  9  ...  (222). 


If  this  equation  be  compared  with  equation  (214),  it  will 
be  found  to  agree  with  it,  mutatis  mutandis,  except  that 
the  co-efficient  2a  is  in  that  equation  2.  This  difference 
manifestly  results  from  the  approximate  character  of  the 
theorem  of  Poncelet. 

Substituting  the  latter  co-efficient  for  the  former,  multiply- 

p  (3 

ing  both  sides  of  the  equation  by  (1 — — fsin.  9),  neglecting 

AI 

terms  of  more  than  two  dimensions  in  — f,  — ,  and  sin.  9,  and 
reducing, 


which  is  the  relation  between  the  moving  and  working 
pressures  in  the  state  bordering  upon  motion.  From  this 
relation  we  obtain  for  the  MODULUS  of  the  band  (equation 
121) 

' 


If  the  angle  0a  be  conceived  to  increase  until  it  exceed 
x,  P,  will  pass  to  the  opposite  side  of  C,0a,  and  (3t  will 
become  negative;  whence  it  is  apparent,  that  equation 


THE   BAND.  225 

(224)  agrees  with  equation  (214)  in  other  respects,  and  in 
the  condition  of  the  ambiguous  sign.  It  is  moreover  appa- 
rent, from  the  form  assumed  by  the  modulus  in  this  case 
and  in  that  of  the  preceding  article,  that  the  greatest 
economy  of  power  is  obtained  by  applying  the  moving  and 
the  working  pressures  on  the  same  side  of  the  line  C^  join- 
ing the  axes  of  the  drums.  This  is  in  fact  but  a  particular 
case  of  the  general  principle  established  in  Art.  168. 


195.  The  initial  tension  T  of  the  band  may  be  deter- 
mined precisely  as  in  the  former  case  (equation  217). 
Representing  by  0  the  angle  sub- 
tended by  the  circumference  which 
the  band  embraces  on  the  second, 
or  driven  drum,  by  P  the  maxi-- 
mum  resistance  opposed  to  its  mo** 
tion  at  the  distance  a,  by  $  t\& 
limiting  angle  of  resistance  between 
the  band  and  the  surface  of  the 
drum,  and  by  ^  and  tt  the  tensions 
upon  the  two  parts  of  the  band, 
when  its  maximum  resistance  being  opposed,  it  is  upon  the 
point  of  slipping;  observing,  moreover,  that  in  this  case 

e  tan.  $ 

2ft— £,)  or  2£  is  represented  (Art.  193.)  by  2T€^~  ^;  then 

e    +"   1 

substituting  in  the  second  of  equations  (220)  this  value  for 
2£,  and  P  and  a  for  Pa  and  #„  and  neglecting  the  exceed- 
ingly small  term  which  involves  the  product  sin.  a,  sin.  9, 
we  have 

(0tan.  $     ^ 
e-6^1T  Va=-p9{2aTco8.ai-P/3a+W2ra}  sin  9. 
e     +     I/ 

Also,  since  aa  represents  the  inclination  of  the  two  parts  of 
the  band  to  one  another ;  since,  moreover,  these  touch  the 
surfaces  of  the  drums,  and  that  6  represents  the  inclination 
of  the  radii  drawn  from  the  centre  of  the  lesser  drum  to  the 
touching  points,  therefore  O=TC— ar  Substituting  this  value 
of  0  in  the  above  equation,  and  solving  it  in  respect  to  T,  we 
have 

15 


226  raE   BAND. 

P(<3— p2/32  sin.  </>)-f  W2p272  sin.  0 

((TT-OI)  tan.  4>\    7*a  — pa  a  COS,  a    sin. 
e    -    l] 
(Tr-aO  tan.  <|>   I 


(225), 


196.  The  modulus  of  the  hand  when  the  two  parts  of  it, 
which  intervene  between  the  drums,  are  made  to  cross  one 
another. 

If  the  directions  of  the  two  parts  of  the  band  be  made 
to  cross,  as  shown  in  the  accompanying 
figure,  the  moving  pressure  Tj  upon  the 
second  drum  is  applied  to  it  on  the  side 
opposite  to  that  on  which  it  is  applied 
when  the  bands  do  not  cross ;  so  that  in 
this  case,  in  order  that  the  greatest  eco- 
nomy of  power  may  be  attained  (Art. 
168.),  the  working  pressure  or  resistance 
P2  should  be  applied  to  it  on  the  side 
opposite  to  that  in  which  it  was  applied 
in  the  other  case,  and  therefore  on  the  side  of  the  line  CjCg, 
opposite  to  that  on  which  the  moving  pressure  Pt  upon  the 
first  drum  is  applied.  This  disposition  of  the  moving  and 
working  pressures  being  supposed,  and  this  case  being  inves- 
tigated by  the  same  steps  as  the  preceding,  we  shall  arrive 
at  precisely  the  same  expressions  (equations  223  and  224) 
for  the  relation  of  the  moving  and  the  working  pressures, 
and  for  the  modulus. 

In  estimating  the  value  of  the  initial  tension  T  (equation 
225)  it  will,  however,  be  found,  that  the  angle  d,  subtended 
at  the  centre  C2  of  the  second  drum  by  the  arc  KML,  which 
is  embraced  by  the  band,  is  no  longer  in  this  case  repre- 
sented by  if— a,  but  by  *  +  «,.  This  will  be  evident  if  we 
consider  that  the  four  angles  of  the  quadrilateral  figure 
C2KIL  being  equal  to  four  right  angles,  and  its  angles  at  K 
and  L  being  right  angles,  the  remaining  angles  KIL  and 
KC2L  are  equal  to  twro  right  angles,  so  that  KC2L=tf— at ; 
but  the  angle  subtended  by  KML  equals  2^— KC2L;  it 
equals  therefore  *  +  «,.  If  this  value  be  substituted  for  t—at 
in  equation  (225)  it  becomes 


THE    TEETH    OF    WHEELS.  227 

If        P(a-pA  sin.  <p)+W2p2  sin.  <p 


2 


A"'  *1\ 

I  ~, \ £ —  tTn — PO  ot  cos.  &  sin 

I      CTT  +  CI;  tan.  (p       »•  >       i^ 

V      +     i/ 


(226.) 


Now  the  fraction  in  the  denominator  of  this  expression 
being  essentially  greater  in  value  than  that  in  the  denomi- 
nator of  the  preceding  (equation  225),  it  follows  that  the 
initial  tension  T,  which  must  be  given  to  the  band  in  order 
that  it  may  transmit  the  work  from  the  one  drum  to  the 
other  under  a  given  resistance  P,  is  less  when  the  two  parts 
of  the  band  cross  than  when  they  do  not,  and,  therefore,  that 
the  modulus  (equation  224)  is  less;  so  that  the  band  is 
worked  with  the  greatest  economy  of  power  (other  things 
'being  the  same)  when  the  two  parts  of  it  which  intervene 
between  the  drums  are  made  to  cross  one  another.  Indeed  it 
is  evident,  that  since  in  this  case  the  arc  embraced  by  the 
band  on  each  drum  subtends  a  greater  angle  than  in  the 
other  case,  a  less  tension  of  the  band  in  this  case  than  in  the 
other  is  required  (Art.  185.)  to  prevent  it  from  slipping 
under  a  given  resistance,  so  that  the  friction  upon  the  axis 
of  the  drums  which  results  from  the  tension  of  the  band  is 
less  in  this  case  than  the  other,  and  therefore  the  work 
expended  on  that  friction  less  in  the  same  proportion. 


THE  TEETH  OF  WHEELS. 

197.  Let  A,  B  represent  two  circles  in  contact  at  D,  and 
moveable  about  fixed  centres  at  C,  and  C2.  It 
is  evident  that  if  by  reason  of  the  friction  of 
these  two  circles  upon  one  another  at  D  any 
motion  of  rotation  given  to  A  be  communicated 
to  B,  the  angles  PC^D  and  QC2D  described  in 
the  same  time  by  these  two  circles,  will  be  such 
as  will  make  the  arcs  PD  and  QD  which  they 
subtend  at  the  circumferences  of  the  circles  equal  to  one 
another.  Let  the  angle  PCjD*  be  represented  by  d,,  and  the 
angle  QC2D  by  d2;  also  let  the  radii  C^D  and  C2D  of  the  cir- 
cles be  represented  by  ?\  and  r9.  Now,  arc  PD=r^^  arc 
r&;  and  since  PD=QD,  therefore  ^A=/1A  j 

*  Or  rather  the  arc  which  this  angle  subtends  to  radius  unity. 


228  THE   TEETH   OF   WHEELS. 


(227). 


The  angles  described,  in  the  same  time,  by  two  circles 
which  revolve  in  contact  are  therefore  inversely  proportional 
to  the  radii  of  the  circles,  so  that  their  angular  velocities 
(Art.  74.)  bear  a  constant  proportion  to  one  another  ;  and  if 
one  revolves  uniformly,  then  the  other  revolves  uniformly  ; 
if  the  angular  revolution  of  the  one  varies  in  any  proportion, 
then  that  of  the  other  varies  in  like  proportion. 

When  the  resistance  opposed  to  the  rotation  of  the  driven 
circle  or  wheel  B  is  considerable,  it  is  no  longer  possible  to 
give  motion  to  that  circle  by  the  friction  on  its  circum- 
ference of  the  driving  circle.  It  becomes  therefore  neces- 
sary in  the  great  majority  of  cases  to  cause  the  rotation  of 
the  driven  wheel  by  some  other  means  than  the  friction  of 
the  circumference  of  the  driving  wheel. 

One  expedient  is  the  band  already  described,  by  means  of 
which  the  weels  may  be  made  to  drive  one  another  at  any 
distances  of  their  centres,  and  under  a  far  greater  resistance 
than  they  could  by  their  mutual  contact.  When,  however, 
the  pressure  is  considerable,  and  the  wheels  may  be  brought 
into  actual  contact,  the  common  and  the  more  certain 
method  is  to  transfer  the  motion 
from  one  to  the  other  by  means  of 
projections  on  the  one  wneel  called 
TEETH,  which  engage  in  similar  pro- 
jections on  the  other. 

In  the  construction  of  these  teeth 
the  problem  to  be  solved  is,  to  give 
such  shapes  to  their  surfaces  of  mu- 
tual contact,  as  that  the  wheels  shall 
be  made  to  turn  by  the  intervention 
of  their  teeth  precisely  as  they  would  by  the  friction  of 
their  circumferences. 

198.  That  it  is  possible  to  construct  teeth  which  shall 
answer  this  condition  may  thus  be  shown. 
Let  mn  and  m'n'  be  two  curves,  the  one 
described  on  the  plane  of  the  circle  A,  and 
the  other  on.  the  plane  of  the  circle  B  ;  and 
let  them  be  such  that  as  the  circle  A  re- 
volves, carrying  round  with  it  the  circle  B, 
by  their  mutual  contact  at  D,  these  two 
curves  mn  and  m'n'  may  continually  touch 


THE   TEETH    OF   WHEELS.  229 

one  another,  altering  of  course,  as  they  will  do  continually, 
their  relative  positions  and  their  point  of  contact  T. 

It  is  evident  that  the  two  circles  would  be  made  to 
revolve  by  the  contact  of  teeth  whose  edges  were  of  the 
forms  of  these  two  curves  mn  and  m'nr  precisely  as  they 
would  by  their  friction  upon  the  circumferences  of  one 
another  at  the  point  D;  for  in  the  former  case  a  certain 
series  of  points  of  contact  of  the  circles  (infinitely  near  to 
one  another)  at  D,  brings  about  another  given  series  of  points 
of  contact  (infinitely  near  to  one  another)  of  the  curves  mn 
and  m'n'  at  T ;  and  in  the  latter  case  the  same  series  of 
points  in  the  curves  mn  and  m'n'  brought  into  contact  neces- 
sarily produces  the  contact  of  the  same  series  of  points  in 
the  two  circumferences  of  the  two  circles  at  D. 


To  construct  teeth  whose  surfaces  of  contact  shall  possess 
the  properties  here  assigned  to  the  curves  mn  and  m'n'  is 
the  problem  to  be  solved.  Of  the  solution  of  this  problem 
the  following  is  the  fundamental  principle : 


199.  In  order  that  two  circles  A  and  B  may  "be  made  to 
revolve  by  the  contact  of  the  surfaces  mn  and  m'n'  of  their 
teeth,  precisely  as  they  would  ly  the  friction  of  their  cir- 
cumferences, it  is  necessary,  and  it  is  suf- 
ficient, that  a  line  drawn  from  the  point 
of  contact  T  of  the  teeth  to  the  point  of 
contact  D  of  the  circumferences  should,  in 
every  position  of  the  point  T,  he  perpendi- 
cular to  the  surfaces  in  contact  there,  i.  e., 
a  normal  to  loth  the  curves  mn  and  m'n'. 

To  prove  this  principle,  we  must  first  establish  the  follow- 
ing LEMMA  : — If  two  circles  M  and  N  be  made  to  revolve 
about  the  fixed  centres  E  and  F  by  their  mu- 
tual contact  at  L,  and  if  the  planes  of  these 
circles  be  conceived  to  be  carried  round  with 
them  in  this  revolution,  and  a  point  P  on  the 
plane  of  M  to  trace  out  a  curve  PQ  on  the 
plane  of  N"  whilst  thus  revolving,  then  is  this 
curved  line  PQ  precisely  the  same  as  would 
have  been  described  on  the  plane  of  N  by  the  same  point  P, 
if  the  latter  plane,  instead  of  revolving,  had  remained  at 
rest,  and  the  centre  E  of  the  circle  M  having  been  released 


230  THE   TEETH    OF   WHEELS. 

from  its  axis,  that  circle  had  been  made  to. roll  (carrying  its 
plane  with  it)  on  the  circumference  of  N.  For  conceive  O 
to  represent  a  third  plane  on  which  the  centres  of  E  and  F 
are  fixed.  It  is  evident  that  if,  whilst  the  circles  M  and  K 
are  revolving  by  their  mutual  contact,  the  plane  O,  to  which 
their  centres  are  both  fixed,  be  in  any  way  moved,  no  change 
will  thereby  be  produced  in  form  of  the  curve  PQ,  which  the 
point  P  in  the  plane  of  M  is  describing  upon  the  plane  of  !N", 
such  a  motion  being  common  to  both  the  planes  M  and  E".* 
Now  let  the  direction  in  which  the  circle  N  is  revolving  be 
that  shown  by  the  arrow,  and  its  angular  velocity  uniform ; 
and  conceive  the  plane  O  to  be  made  to  revolve  about  F  with 
an  angular  velocity  (Art.  74)  which  is  equal  to  that  of  N, 
but  in  an  opposite  direction,  communicating 
this  angular  velocity  to  M  and  !N".  these  re- 
volving meantime  in  respect  to  one  another, 
and  by  their  mutual  contact,  precisely  as  they 
did  before.f 

It  is  clear  that  the  circle  E"  being  carried 
round  by  its  own  proper  motion  in  one  direc- 
tion, and  by  the  motion  common  to  it  and  the  plane  O  with 
the  same  angular  velocity  in  the  opposite  direction,  will,  in 
reality  rest  in  space ;  whilst  the  centre  E  of  the  circle  M, 
having  no  motion  proper  to  itself,  will  revolve  with  the 
angular  velocity  of  the  plane  O,  and  the  various  other  points 
in  that  circle  with  angular  velocities,  compounded  of  their 
proper  velocities,  and  those  which  they  receive  in  common 
with  the  plane  O,  these  velocities  neutralising  one  another 
at  the  point  L  of  the  circle,  by  which  point  it  is  in  contact 
with  the  circle  N".  So  that  whilst  .M  revolves  round  1ST,  the 
point  L,  by  which  the  former  circle  at  any  time  touches  the 
other,  is  at  rest ;  this  quiescent  point  of  the  circle  M  never- 
theless continually  varying  its  position  on  the  circumferences 
of  both  circles,  and  the  circle  M  being  in  fact  made  to  roll 
on  the  circle  N  at  rest. 

Thus,  then,  it  appears,  that  by  communicating  a  certain 
common  angular  velocity  to  both  the  circles  M  and  !N"  about 


*  Thus  for  instance,  if  the  circles  M  and  N  continue  to  revolve,  we  may 
evidently  place  the  whole  machine  in  a  ship  under  sail,  in  a  moving  carriage, 
or  upon  a  revolving  wheel,  without  in  the  least  altering  the  form  of  the  curve, 
which  the  point  P,  revolving  with  the  plane  of  the  circle  M,  is  made  to  trace 
on  the  plane  of  N,  because  the  motion  we  have  communicated  is  common  to 
both  these  circles. 

f  M  and  N  may  be  imagined  to  be  placed  upon  a  horizontal  wheel  0,  first  at 
rest,  and  then  made  to  revolve  backwards  in  respect  to  the  motion  of  N. 


THE    TEETH    OF    WHEELS.  231 

the  centre  F,  the  former  circle  is  made  to  Toll  upon  the  other 
at  rest ;  and,  moreover,  that  this  common  angular  velocity 
does  not  alter  the  form  of  the  curve  PQ,  which  a  point  P  in 
the  plane  of  the  one  circle  is  made  to  trace  upon  the  plane 
of  the  other,  or,  in  other  words,  that  the  curve  traced  under 
these  circumstances  is  the  same,  whether  the  circles  revolve 
round  fixed  centres  by  their  mutual  contact,  or  whether  the 
centre  of  one  circle  be  released,  and  it  be  made  to  roll  upon 
the  circumference  of  the  other  at  rest. 

This  lemma  being  established,  the  truth  of  the  proposition 
stated  at  the  head  of  this  article  becomes  evident ;  for  if  M 
roll  on  the  circumference  of  N,  it  is  evident  that  P  will,  at 
any  instant,  be  describing  a  circle  about  their  point  of  con- 
tact L.* 

Since  then  P  is  describing,  at  every  instant,  a  circle  about 
L  when  M  rolls  upon  N,  !N"  being  fixed,  and  since  the  curve 
described  by  P  upon  this  supposition  is  precisely  the  same 
as  would  have  been  traced  by  it  if  the  centres  of  both  cir- 
cles had  been  fixed,  and  they  had  turned  by  their  mutual 
contact,  it  follows  that  in  this  last  case  (when  the  circles 
revolve  about  fixed  centres  by  their  mutual  contact)  the 
point  P  is  at  any  instant  of  the  revolution  describing,  during 
that  instant,  an  exceedingly  small  circular  arc  about  the 
point  L ;  whence  it  follows  that  PL  is  always  a  perpendicu- 
lar to  the  cur,ve  PQ  at  the  point  P,  or  a  normal  to  it. 

ISTow  let  p  be  a  point  exceedingly  near  to  T  in  the  curve 
raW,  which  curve  is  fixed  upon  the  plane 
of  the  circle  A.  It  is  evident  that,  as  the 
point  p  passes  through  its  contact  with  the 
curve  mn  at  T  (see  Art.  195.),  it  will  be 
made  to  describe,  on  the  plane  of  the  circle 
B,  an  exceedingly  small  portion  of  that 
curve  TTiTi.  But  the  curve  which  it  is 
(under  these  circumstances)  at  any  instant 
describing  upon  the  plane  of  B  has  been  shown  to  be 
always  perpendicular  to  the  line  DT ;  the  curve  mn  is  there- 
fore at  the  point  T  perpendicular  to  the  line  DT ;  whence  it 
follows  that  the  curve  m'n'  is  also  perpendicular  to  that  line, 
and  that  DT  is  a  normal  to  ~both  those  curves  at  T.  This  is 
the  characteristic  property  of  the  curves  mn  and  raW,  so  that 
they  may  satisfy  the  condition  of  a  continual  contact  with 


*  For  either  circle  may  be  imagined  to  be  a  polygon  of  an  infinite  number 
of  sides,  on  one  of  the  angles  of  which  the  rolling  circle  will,  at  any  instant, 
be  in  the  act  of  turning. 


232  THE   TEETH    OF    WHEELS. 

one  another,  whilst  the  circles  revolve  by  the  contact  of 
their  circumferences  at  D,  and  therefore  conversely,  so  that 
these  curves  may,  by  their  mutual  contact,  give  to  the  cir- 
cles the  same  motion  as  they  would  receive  from  the  contact 
of  their  circumferences. 


200.  To  describe,  ~by  means  of  circular  arcs,  the  form  of  a 
tooth  on  one  wheel  which  shall  work  truly  with  a  tooth  of 
any  given  form  on  another  wheel. 

Let  the  wheels  be  required  to  revolve  by  the  action  of 
their  teeth,  as  they  would  by  the 
contact  of  the  circles  ABE  and 
ADF,  called  their  primitive  ov  pitch 
circles.  Let  AB  represent  an  arc 
of  the  pitch  circle  ABE,  included 
between  any  two  similar  points  A. 
and  B  of  consecutive  teeth,  and  let 
AD  represent  an  arc  of  the  pitch 
circle  ADF  equal  to  the  arc  AB,  so 
that  the  points  D  and  B  may  come 
simultaneously  to  A,  when  the  cir- 
cles are  made  to  revolve  by  their 
mutual  contact.  AB  and  AD  are 
called  the  pitches  of  the  teeth  of  the  two  wheels.  Divide 
each  of  these  pitches  into  the  same  number  of  equal  parts 
in  the  points  a,  b,  &c.,  a',  b',  &c. ;  the  points  a  and  a',  b  and 
b',  &c.,  will  then  be  brought  simultaneously  to  the  point  A. 
Let  mn  represent  the  form  of  the  face  of  a  tooth  on  the 
wheel,  whose  centre  is  C,,  with  which  tooth  a  corresponding 
tooth  on  the  other  wheel  is  to  work  truly ;  that  is  to  say, 
the  tooth  on  the  other  wheel,  whose  centre  is  C2,  is  to  be  cut, 
so  that,  driving  the  surface  mn*  or  being  driven  by  it,  the 
wheels  shall  revolve  precisely  as  they  would  by  the  con- 
tact of  their  pitch  circles  ABE  and  ADF  at  A.  From  A 
measure  the  least  distance  A«  to  the  curve  mn,  and  with 
radius  A«  and  centre  A  describe  a  circular  arc  «/3  on  the 
plane  of  the  circle  whose  centre  is  Ca.  From  a  measure,  in 
like  manner,  the  least  distance  a*',  to  the  curve  mn,  and 
with  this  distance  a*'  and  the  centre  a,  describe  a  circular 
arc  £7,  intersecting  the  arc  a/3  in  ft.  From  the  point  b 
measure  similarly  the  shortest  distance  W  to  mn,  and  with 


THE   TEETH    OF   WHEELS.  233 

the  centre  V  and  this  distance  Wl  describe  a  circular  arc 
7(?,  intersecting  £7  in  7,  and  so  with  the  other  points  of 
division.  A  curve  touching  these  circular  arcs  a/3,  ^7,  7^ 
&c.,  will  give  the  true  surface  or  boundary  of  the  tooth.* 

In  order  to  prove  this  let  it  be  observed,  that  the  shortest 
distance  afl!  from  a  given  point  a  to  a  given  curve  mn  is  a 
normal  to  the  curve  at  the  point  a'  in  which  it  meets  it ;  and 
therefore,  that  if  a  circle  be  struck  from  this  point  a  with  this 
least  distance  as  a  radius,  then  this  circle  must  touch  the 
curve  in  the  point  «',  and  the  curve  and  circle  have  a  com- 
mon normal  in  that  point. 

Now  the  points  a  and  a  will  be  brought  by  the  revolution 
of  the  pitch  circles  simultaneously  to  the  point  of  contact  A, 
and  the  least  distance  of  the  curve  mn  from  the  point  A  will 
then  be  #a',  so  that  the  arc  ^7  will  then  be  an  arc  struck 
from  the  centre  A,  with  this  last  distance  for  its  radius.  This 
circular  arc  ^7  will  therefore  touch  the  curve  mn  in  the  point 
a'  and  the  line  aa',  which  will  then  be  a  line  drawn  from 
the  point  of  contact  A  of  the  two  pitch  circles  to  the  point 
of  contact  «'  of  the  two  curves  mn  and  m'n ',  will  also  be  a 
normal  to  both  curves  at  that  point.  The  circles  will  there- 
fore at  that  instant  drive  one  another  (Art.  196.)  by  the  con- 
tact of  the  surfaces  mn  and  m'n' ,  precisely  as  they  would  by 
the  contact  of  their  circumferences.  And  as  every  circular 
arc  of  the  curve  m'n  similar  to  ^7  becomes  in  its  turn  the 
acting  surface  of  the  tooth,  it  will,  in  like  manner,  at  one 
point  work  truly  with  a  corresponding  point  of  mn,  so  that 
the  circles  will  thus  drive  one  another  truly  at  as  many 
points  of  the  surfaces  of  their  teeth,  as  there  have  been  taken 
points  of  division  a,  5,  &c.  and  arcs  a/3,  £7.  &c.f 

*  This  method  of  describing,  geometrically,  the  forms  of  teeth  is  given,  without 
demonstration,  by  M.  Poncelet  in  his  Mecanique  Industrielle,  3me  partie,  Art.  60. 

f  The  greater  the  number  of  these  points  of  division,  the  more  accurate  the 
form  of  the  tooth.  It  appears,  however,  to  be  sufficient 
in  most  cases,  to  take  three  points  of  division,  or  even 
two,  where  no  great  accuracy  is  required.  M.  Poncelet 
(Mee.  Indust.  3me  partie,  Art.  60.)  has  given  the  following, 
yet  easier,  method  by  which  the  true  form  of  the  tooth 
may  be  approximated  to  with  sufficient  accuracy  in  most 
cases.  Suppose  the  given  tooth  N  upon  the  one  wheel  to 
be  placed  in  the  position  in  which  it  is  first  to  engage  or 
disengage  from  the  required  tooth  on  the  other  wheel, 
and  let  Aa  and  A6  be  equal  arcs  of  the  pitch  circles  of 
the  two  wheels  whose  point  of  contact  is  A.  Draw  Aa 
the  shortest  distance  between  A  and  the  face  of  the  tooth 
N ;  join  aa;  bisect  that  line  in  m,  and  draw  mn  perpendi- 
cular to  aa  intersecting  the  circumference  Aa  in  n.  If 
from  the  centre,  n  a  circular  arc  be  described  passing 
through  the  points  a  and  a,  it  will  give  the  required  form 
of  the  tooth  nearly. 


234: 


INVOLUTE  TEETH. 


INVOLUTE  TEETH. 

201.  The  teeth  of  two  wheels  will  work  truly  together  if  they 
he  hounded  hy  curves  of  the  form  traced  out  hy  the  extremity 
of  a  flexible  line,  unwinding  from  the  circumference  of  a 
circle,  and  called  the  involute  of  a  circle,  provided  that  the 
circles  of  which  these  are  the  involutes  he  concentric  with 
the  pitch  circles  of  the  wheels,  and  have  their  radii  in  the 
same  proportion  with  the  radii  of  the  pitch  circles. 

Let  OE  and  OF  represent  the  pitch  circles  of  two  wheels, 
AG  and  BH  two  circles  concentric  with 
them  and  having  their  radii  CtA  and  C.,B 
in  the  same  proportion  with  the  radii  CXO 
and  C2O  of  the  pitch  circles.  Also  let  mn 
and  m'n'  represent  the  edges  of  teeth  on  the 
two  wheels  struck  by  the  extremities  of  flexi- 
ble lines  unwinding  from  the  circumferences 
of  the  circles  AG  and  BH  respectively.  Let 
these  teeth  be  in  contact,  in  any  position 
of  the  wheels,  in  the  point  T,  and  from  the 
point  T  draw  TA  and  TB  tangents  to  the 
generating  circles  GA  and  BH  in  the  points 
A  and  B.  Then  does  AT  evidently  represent  the  position  of 
the  flexible  line  when  its  extremity  was  in  the  act  of  gene- 
rating the  point  T  in  the  curve  mn  /  wrhence  it  follows,  that 
AT  is  a  normal  to  the  curve  mn  at  the  point  T* ;  and  in 
like  manner  that  BT  is  a  normal  to  the  curve  m'n'  at  the 
same  point  T.  Now  the  two  curves  have  a 
common  tangent  at  T;  therefore  their  nor- 
mals TA  and  TB  at  that  point  are  in  the  same 
straight  line,  being  both  perpendicular  to  their 
tangent  there.  Since  then  ATB  is  a  straight 
line,  and  that  the  vertical  angles  at  the  point 
o  where  AB  and  CjC,  intersect  are  equal,  as 
also  the  right  angles  at  A  and  B,  it  follows 
that  the  triangles  AoC^and  B<?02  are  similar, 
and  that  0,o  :  Cto  ::  C,A  :  C2B.  But  C,A  : 
CQB  ::  0,0  :  C2O  ;  .-.  Cto  :  0,0  ::  0,0: 
C2O  ;  therefore  the  points  O  and  o  coincide, 
and  the  straight  line  AB,  which  passes  through  the  point  of 

*  For  if  the  circle  be  conceived  a  polygon  of  an  infinite  number  of  sides,  it 
is  evident  that  the  line,  when  in  the  act  of  unwinding  from  it  at  A,  is  turning 
upon  one  of  the  angles  of  that  polygon,  and  therefore  that  its  extremity  is, 
through  au  infinitely  small  angle,  describing  a  circular  arc  about  that  point. 


INVOLUTE   TEETH.  235 

contact  T  of  the  two  teeth,  and  is  perpendicular  to  the  sur- 
faces of  both  at  that  point,  passes  also  through  the  point  of 
contact  O  of  the  pitch  circles  of  the  wheels.  Now  this  is 
true,  whatever  be  the  positions  of  the  wheels,  and  whatever, 
therefore,  be  the  points  of  contact  of  the  teeth.  Tims  then 
the  condition  established  in  Art.  199.  as  that  necessary  and 
sufficient  to  the  true  action  of  the  teeth  of  wheels,  viz.  "  that 
a  line  drawn  from  the  point  of  contact  to  the  pitch  circles  tc 
the  point  of  contact  of  the  teeth  should  be  a  normal  to  their 
surfaces  at  that  point,  in  all  the  different  positions  of  the 
teeth,"  obtains  in  regard  to  involute  teeth.* 

The  point  of  contact  T  of  the  teeth  moves  along  the  straight- 
line  AB,  which  is  drawn  touching  the  generating  circles  BEL 
and  AG  of  the  involutes  ;  this  line  is  what  is  called  the  locus 
of  the  different  points  of  contact.  Moreover,  this  property 
obtains,  whatever  may  be  the  number  of  teeth  in  contact  at 
once,  so  that  all  the  points  of  contact  of  the  teeth,  if  there 
be  more  than  one  tooth  in  contact  at  once,  lie  always  in  this 


line ;  which  is  a  characteristic,  and  a  most  important  pro- 
perty of  teeth  of  the  involute  form.     Thus  in  the  above 

*  The  author  proposes  the  following  illustration  of  the  action  of  involute 
teeth,  which  he  believes  to  be  new.  Conceive  AB  to  represent  a  band  passing 
round  the  circles  AG  and  BH,  the  wheels  would  evidently  be  driven  by  this 
band  precisely  as  they  would  by  the  contact  of  their  pitch  circles,  since  the 
radii  of  AG  and  BH  are  to  one  another  as  the  radii  of  the  pitch  circles.  Con- 
ceive, moreover,  that  the  circles  BH  and  AG  carry  round  with  them  their 
planes  as  they  revolve,  and  that  a  tracer  is  fixed  at  any  point  T  of  the  band, 
tracing,  at  the  same  time,  lines  mn  and  m'n',  upon  both  planes,  as  they  revolve 
beneath  it.  It  is  evident  that  these  curves,  being  traced  by  the  same  point, 
must  be  in  contact  in  all  positions  of  the  circles  when  driven  by  the  band,  and 
therefore  when  driven  by  their  mutual  contact.  The  wheels  would  therefore 
be  driven  by  the  contact  of  teeth  of  the  forms  mn  and  m'n'  thus  traced  by  the 
point  T  of  the  band  precisely  as  they  would  by  the  contact  of  their  pitch  cir- 
cles. NoV  it  is  easily  seen,  that  the  curves  mn  and  m'n',  thus  described  by  the 
point  T  of  the  band,  are  involutes  of  the  circles  AG  and  BH. 


236  EPICYCLOIDAL   AND   HYPCCYCLOIDAL   TEETH. 

figure,  which  represents  part  of  two  wheels  with  involute 
teeth,  it  will  be  seen  that  the  points  r  s  of  contact  of  the 
teeth  are  in  the  same  straight  line  touching  the  base*  of  one 
of  the  involutes,  and  passing  through  the  point  of  contact  A 
of  the  pitch  circles,  as  also  the  points  A  and  J  in  that  touch- 
ing the  base  of  the  other. 


EPICYCLOIDAL  AND  HYPOOYCLOIDAL  TEETH. 

202.  If  one  circle  be  made  to  roll  externally  on  the  cir- 
cumference of  another,  and  if,  whilst  this  mo- 
tion is  taking  place,  a  point  in  the  circumfe- 
rence of  the  rolling  circle  be  made  to  trace 
out  a  curve  upon  the  plane  of  the  fixed  circle, 
the  curve  so  generated  is  called  an  EPICYCLOID, 
the  rolling  circle  being  called  the  generating 
circle  of  the  epicycloid,  and  the  circle  upon 
which  it  rolls  its  oase. 

If  the  generating  circle,  instead  of  rolling 
on  the  outside  or  convex  circumference  of  its 
base,  roll  on  its  inside  or  concave  circumfe- 
rence, the  curve  generated  is  called  the  HYPOCYCLOID. 

Let  PQ  and  Pit  be  respectively  an  epicycloid  and  a  hypo- 
cycloid,  having  the  same  generating  circle  APH,  and 
having  for  their  bases  the  pitch  circles  AF  and  AE  of  two 
wheels.  If  teeth  be  cut  upon  these  wheels,  whose  edges 
coincide  with  the  curves  PQ  and  PR,  they  will  work  truly 
with  one  another ;  for  let  them  be  in  contact  at  P,  and  let 
their  common  generating  circle  APH  be  placed  so  as  to 
touch  the  pitch  circles  of  both  wheels  at  A ,  then  will  its  cir- 
cumference evidently  pass  through  the  point  of  contact  P 
of  the  teeth :  for  if  it  be  made  to  roll  through  an  exceed- 
ingly small  angle  upon  the  point  A,  rolling  there  upon  the 
circumference  of  loth  circles,  its  generating  point  will 
traverse  exceedingly  small  portions  of  both  curves;  since 
then  a  given  point  in  the  circumference  of  the  circle  APII 
is  thus  shown  to  be  at  one  and  the  same  time  in  th£  perime- 
ters of  both  the  curves  PQ  and  PR,  that  point  must  of 
necessity  be  the  point  of  contact  P  of  the  curves ;  since, 


*  The  circles  from  which  the  involutes  are   described  are  called  their 
This  cut  and  that  at  page  237.  are  copied  from  Mr.  Hawkins'  edition  of  Camus 
on  the  Teeth  of  Wheels. 


EPICYCLOIDAL   AND   HYPOCYCLOIDAL    TEETH. 


237 


moreover,  when  the  circle  APH  rolls  upon  the  point  A,  its 
generating  point  traverses  a  small  portion  of  the  perimeter 
of  each  of  the  curves  PQ  and  PR  at  P,  it  follows  that  the 
line  AP  is  a  normal  to  both  curves  at  that  point ;  for  whilst 
the  circle  APH  is  rolling  through  an  exceedingly  small 
angle  upon  A,  the  point  P  in  it,  is  describing  a  circle  about 
that  point  whose  radius  is  AP.*  Teeth,  therefore,  whose 
edges  are  of  the  forms  PQ  and  PR  satisfy  the  condition 
that  the  line  AP  drawn  from  the  point  of  contact  of  the 
pitch  circles  to  any  point  of  contact  of  the  teeth  is  a  normal 
to  the  surfaces  of  both  at  that  point,  which  condition  has  been 
shown  (Art.  199.)  to  be  that  necessary  and  sufficient  to  the 
correct  working  of  the  teeth.f 
Thus  then  it  appears,  that  if  an  epicycloid  be  described 


*  The  circle  AFH  may  be  considered  a  polygon  of  an  infinite  number  of 
sides,  on  one  of  the  angles  of  which  polygon  it  may  at  any  instant  be  con- 
ceived to  be  turning. 

f  The  entire  demonstration  by  which  it  has  been  here  shown  that  the 
curves  generated  by  a  point  in  the  circumference  of  a  given  generating  circle 
APH  rolling  upon  the  convex  circumference  of  one  of  the  pitch  circles,  and 
upon  the  concave  circumference  of  the  other  are  proper  to  form  the  edges  of 
contact  of  the  teeth,  is  evidently  applicable  if  any  other  generating  curve  be 
substituted  for  APH.  It  may  be  shown  precisely  in  the  same  manner,  that 
the  curves  PQ  and  PR  generated  by  the  rolling  of  any  such  curve  (not  being 
a  circle)  upon  the  pitch  circles,  possess  this  property,  that  the  line  PA  drawn 
from  any  point  of  their  contact  to  the  point  of  contact  of  their  pitch  circles 
is  a  normal  to  both,  which  property  is  necessary  and  sufficient  to  their  correct 
action  as  teeth.  This  was  first  demonstrated  as  a  general  principle  of  the  con- 
struction of  the  teeth  of  wheels  by  Mr.  Airy,  in  the  Cambridge  Phil.  Trans, 
vol.  ii.  He  has  farther  shown,  that  a  tooth  of  any  form  whatever  being  cut 
upon  a  wheel,  it  is  possible  to  find  a  curve  which,  rolling  upon  the  pitch  circle 
of  that  wheel,  shall  by  a  certain  generating  point  traverse  the  edge  of  the 
given  tooth.  The  curve  thus  found  being  made  to  roll  on  the  circumference 
of  the  pitch  circle  of  a  second  wheel,  will  therefore  trace  out  the  form  of  a 
tooth  which  will  work  truly  with  the  first.  This  beautiful  property  involves 


238  EPICYCLOID AL   AND   HYPOCYCLOIDAL   TEETH. 

on  the  plane  of  one  of  the  wheels  with  any  generating 
circle,  and  with  the  pitch  circle  of  that  wheel  for  its  base  ; 
and  if  a  hypocycloid  be  described  on  the  plane  of  the  other 
wheel  with  the  pitch  circle  of  that  wheel  for  its  base ;  and 
if  the  faces  or  acting  surfaces  of  the  teeth  on  the  two 
weeels  be  cut  so  as  to  coincide  with  this  epicycloid  and  this 
hypocycloid  respectively,  then  will  the  wheels  be  driven 
correctly  by  the  intervention  of  these  teeth.  Parts  of  two 
wheels  having  epicycloidal  teeth  are  represented  in  the  pre- 
ceding figure. 


203.  LEMMA.  —  If  the  diameter  of  the  generating  circle  of  a 
hypocycloid  equal  the  radius  of  its  ~base,  the  hypocycloid 
becomes  a  straight  line  having  the  direction  of  a  radius  of 
its  base. 

Let  D  and  d  represent  two  positions  of  the  centre  of  such 
a  generating  circle,  and  suppose  the 
generating  point  to  have  been  at  A  in 
the  first  position,  and  join  AC  ;  then 
will  the  generating  point  be  at  P  in  the 
second  position,  i.  e.  at  the  point  where 
CA  intersects  the-  circle  in  its  second 
position;  for  join  Ca  and  P<#,  then 
/  Yda=  Z  PCW+  /  CP^=2ACa.  Also 


ACa  ;  .'.  arc  A#  =arc  P#.  Since  then  the  arc  aP  equals 
the  arc  «A,  the  point  P  is  that  which  in  the  first  position 
coincided  with  A,  i.  e.  P  is  the  generating  point  ;  and  this 
is  true  for  all  positions  of  the  generating  circle  ;  the  gene- 
rating point  is  therefore  always  in  the  straight  line  AC. 
The  edge,  therefore,  of  a  hypocycloidal  tooth,  the  diameter 
of  whose  generating  circle  equals  half  the  diameter  of  the 
pitch  circle  of  its  wheel,  is  a  straight  line  whose  direction 
is  towards  the  centre  of  the  wheel.* 


the  theoretical  solution  of  the  problem  which  Poncelet  has  solved  by  the 
geometrical  construction  given  to  Article  200.  If  the  rolling  curve  be  a 
logarithmic  spiral,  the  involute  form  of  tooth  will  be  generated. 

*  The  following  very  ingenious  application  has  been  made  of  this  property 
of  the  hypocycloid  to  convert  a  circular  into  an  alternate  rectilinear  motion. 
AB  represents  a  ring  of  metal,  fixed  in  position,  and  having  teeth  cut  upon  ita 


TO    SET   OUT   THE   TEETH   OF   WHEELS. 


To    SET    OUT    THE    TEETH    OF    WHEELS. 

204.  All  the  teeth  of  the  same  wheel  are  constructed  of 
the  same  form  and  of  equal  dimensions :  it  would,  indeed, 
evidently  be  impossible  to  construct  two  wheels  with  dif- 
ferent numbers  of  teeth,  which  should  work  truly  with  one 
another,  if  all  the  teeth  on  each  wheel  were  not  thus  alike. 

All  the  teeth  of  a  wheel  are  therefore  set  out  by  the  work- 
man from  the  same  pattern  or  model,  and  it  is  in  determining 
the  form  and  dimensions  of  this  single  pattern  or  model  of 
one  or  more  teeth  in  reference  to  the  mechanical  effects 
which  the  wheel  is  to  produce,  when  all  its  teeth  are  cut  out 
upon  it  and  it  receives  its  proper  place  in  the  mechanical 
combination  of  which  it  is  to  form  a  part,  that  consists  the 
art  of  the  description  of  the  teeth  of  wheels. 

The  mechanical  function  usually  assigned  to  toothed  wheels 
is  the  transmission  of  work  under  an  increased  or  diminished 
velocity.  If  CD,  DE,  &c.,  represent  arcs  of  the  pitch  circle 

concave  circumference.  C  is  the  centre  of 
a  wheel,  having  teeth  cut  in  its  circum- 
ference to  work  with  those  upon  the  circum- 
ference of  the  ring,  and  having  the  diame- 
ter of  its  pitch  circle  equal  to  half  that  of 
the  pitch  circle  of  the  teeth  of  the  ring. 
This  being  the  case,  it  is  evident,  that  if  the 
pitch  circle  of  the  wheel  C  were  made  to 
roll  upon  that  of  the  ring,  any  point  in  its 
circumference  would  describe  a  straight  line 
passing  through  the  centre  D  of  the  ring; 
but  the  circle  C  would  roll  upon  the  ring  by 
the  mutual  action  of  their  teeth  as  it  wo  ild 
by  the  contact  of  their  pitch  circles ;  if  the 
circle  C  then  be  made  to  roll  upon  the  ring 
by  the  intervention  of  teeth  cut  upon  both,  any  point  in  the  circumference  of 
C  will  describe  a  straight  line  passing  through  D.  Now,  conceive  C  to  be  thus 
made  to  roll  round  the  ring  by  means  of  a  double  or  forked  link  CD,  between 
the  two  branches  of  which  the  wheel  is  received,  being  perforated  at  tlu-ii- 
extremities  by  circular  apertures,  which  serve  as  bearings  to  the  solid  axis  of 
the  wheel.  At  its  other  extremity  D,  this  forked  link  is  rigidly  connected 
with  an  axis  passing  through  the  centre  of  the  ring,  to  which  axis  is  commu- 
nicated the  circular  motion  to  be  converted  by  the  instrument  into  an  alter- 
nating rectilinear  motion.  This  circular  motion  will  thus  be  made  to  carry 
the  centre  C  of  the  wheel  round  the  point  D,  and  at  the  same  time,  cause  it  to 
roll  upon  the  circumference  of  the  ring.  Xow,  conceive  the  axis  C  of  the 
wheel,  which  forms  part  of  the  wheel  itself,  to  be  prolonged  beyond  the  collar 
in  which  it  turns,  and  to  have  rigidly  fixed  upon  its  extremity  a  bar  CP.  It  is 
evident  that  a  point  P  in  this  bar,  whose  distance  from  the  axis  C  of  the  wheel 
equals  the  radius  of  its  pitch  circle,  will  move  precisely  as  a  point  in  the  pitch  circle 
of  the  wheel  moves,  and  therefore  that  it  will  describe  continually  a  straight 
line  passing  through  the  centre  D  of  the  ring.  This  point  P  receives,  there 
fore,  the  alternating  rectilinear  motion  which  it  was  required  to  communicate. 


240 


TO   SET    OUT   THE   TEETH   OF   WHEELS. 


of  a  wheel  intercepted  between  similar  points  of  consecutive 
teeth  (the  chords  of  which  arcs  are  called  the  pitches  of  the 
teeth),  it  is  evident  that  all  these  arcs  must  be  equal,  since 
the  teeth  are  all  equal  and  similarly  placed  ;  so  that  each 
tooth  of  either  wheel,  as  it  passes  through  its  contact  with  a 
corresponding  tooth  of  the  other,  carries  its  pitch  line  through 
the  same  space  CD,  over  the  point  of  contact  C  of  the  pitch 
lines.  Since,  therefore,  the  pitch  line  of  the  one  wheel  is 
carried  over  a  space  equal  to  CD,  and  that  of  the  other  over 
a  space  equal  to  cd  by  the  contact  of  any  two  of  their  teeth, 
and  since  the  wheels  revolve  by  the  contact  of  their  teeth 
as  they  would  by  the  contact  of  their  pitch  circles  at  C,  it 
follows  that  the  arcs  CD  and  cd  are  equal.  Now  let  T\  and 
r2  represent  the  radii  of  the  pitch  circles  of  the  two  wheels, 
then  will  2^  and  2irr9  represent  the  circumferences  of  their 
pitch  circles  ;  and  if  n^  and  n9  represent  the  numbers  of 

teeth  cut  on  them  respectively,  then  CD=  —  l  and  cd  =  —  -2, 


,1  .  . 

therefore,  ---  > 


na 


(227); 


Therefore  the  radii  of  the  pitch  circles  of  the  two  wheels 
must  be  to  one  another  as  the  numbers  of  teeth  to  be  cut 
upon  them  respectively. 

Again,  let  m,  represent  the  number  of  revolutions  made 
by  the  first  wheel,  whilst  m2  revolutions  are  made  by  the 
second ;  then  will  ^irrlml  represent  the  space  described  by 


A   TKAIN    OF    WHEELS.  24:1 

the  circumference  of  the  pitch  circle  of  the  first  wheel  while 
these  revolutions  are  made,  and  27rr2m2  that  described  by  the 
circumference  of  the  pitch  circle  of  the  second;  but  the 
wheels  revolve  as  though  their  pitch  circles  were  in  contact, 
therefore  the  circumferences  of  these  circles  revolve  through 
equal  spaces,  therefore  27ir1m1=2'nTam2  ; 


(228). 


The  radii  of  the  pitch  circles  of  the  wheels  are  therefore 
inversely  as  the  numbers  of  revolutions  made  in  the  same 
time  by  them. 

Equating  the  second  members  of  equations  (227)  and;  (223)) 


(229).. 


The  numbers  of  revolutions  made  by  the  wheel's  in  the  same 
time  are  therefore  to  one  another  inversely  as  the  numbers 
of  teeth. 


205.  In  a  train  of  wheels,  to  determine  how  many  revolutions 
the  last  wheel  makes  whilst  the  first  is  making  any  given 
number  of  revolutions. 

When  a  wheel,  driven  by  another,  carries  its  axis  round 

with  it,  on  which  axis  a  third 
wheel  is  fixed,  engaging  with  and 
giving  motion  to  ^fourth,  which, 
in  like  manner,  is  fixed  upon  its 
axis,  and  carries  round  with  it  a 
fifth  wheel  fixed  upon  the  same 
axis,  which  fifth  wheel  engages 
with  a  sixth  upon  another  axis, 
and  so  on  as  shown  in  the  above  figure,  the  combination 
forms  a  train  of  wheels.  Let  nl9  n»  n»  .  .  .  n^  represent  the 
numbers  of  teeth  in  the  successive  wheels  forming  such  a 
train  of  p  pairs  of  wheels ;  and  whilst  the  first  wheel  is 
making  m  revolutions,  let  the  second  and  third  (which  revolve 
together,  being  fixed  on  the  same  axis)  make  m^  revolutions ; 
the  fourth  and  fifth  (which,  in  like  manner,  revolve  together) 
w2  revolutions,  the  sixth  and  seventh  m^  and  so  on ;  and  let 
the  last  or  2pth  wheel  thus  be  made  to  revolve  mp  times  whilst 

16 


242  A   TKAIN    OF    WHEELS. 

the  first  revolves  m  times.  Then,  since  the  first  wheel  which 
has  nt  teeth  gives  motion  to  the  second  which  has  nt  teeth, 
and  that  whilst  the  former  makes  m  revolutions  the  latter 

makes  m,  revolutions,   therefore  (equation    229),  — i  =  — •  ; 

and  since,  while  thfe  third  wheel  (which  revolves  with  the 
second,  makes  ma  revolutions,  the  fourth  makes  ra2  revolu- 
tions ;  therefore,  — 2  =  — .     Similarly,  since  while   the   fifth 
iml      n4 

wheel,  which  has  n6  teeth,  makes  ma  revolutions  (revolving 
with  the  fourth),  the  sixth,  which  has  n6  teeth,  makes  m,  revo- 
lutions ;  therefore  — -3  =  — .     In  like  manner  — 4  =  — ,  &c.  &c. 
ma     n6  mz      nj 

— .     Multiplying  these  equations  together,    and 

striking  out  factors  common  to  the  numerator  and  denomi- 
nator of  the  first  member  of  the  equation  which  results  from 
their  multiplication,  we  obtain 

mp       nt  .  nz  .  n6  .  .  .  .  nZp-i  f 

—  = —  —  ....  (230). 

m        n^  .  nA  .  n6  .  .  .  .  n^p 

The  factors  in  the  numerator  of  this  fraction  represent  the 
numbers  of  teeth  in  all  the  driving  wheels  of  tfyis  train, 
and  those  in  the  denominator  the  numbers  of  teeth  in  the 
driven  wheels,  or  followers  as  they  are  more  commonly 
called. 

If  the  numbers  of  teeth  in  the  former  be  all  equal  and 
represented  by  n^  and  the  numbers  of  teeth  in  the  latter 
also  equal  and  represented  by  7ia,  then 


>-  =  . (231). 

m        \nz/ 

Having  determined  what  should  be  the  number  of  teeth 
in  each  of  the  wheels  which  enter  into  any  mechanical 
combination,  with  a  reference  to  that  particular  modification 
of  the  velocity  of  the  revolving  parts  of  the  machine,  which 
is  to  be  produced  by  that  wheel,*  it  remains  next  to  consider, 
what  must  be  the  dimensions  of  each  tooth  of  the  wheel,  so 

*  The  reader  is  referred  for  a  more  complete  discussion  of  this  subject  (which 
belongs  more  particularly  to  descriptive  mechanics)  to  Professor  Willis's  Prin- 
ciples of  Mechanism,  chap,  vii.,  or  to  Camus  on  the  Teeth  of  Wheels,  by  Haw- 
kins, p.  90. 


THE  STRENGTH  OF  TEETH.  243 

that  it  may  be  of  sufficient  strength  to  transmit  the  work 
which  is  destined  to  pass  through  it,  under  that  velocity,  or 
to  bear  the  pressure  which  accompanies  the  transmission  of 
that  work  at  that  particular  velocity ;  and  it  remains  further 
to  determine,  what  must  be  the  dimensions  of  the  wheel 
itself  consequent  upon  these  dimensions  of  each  tooth,  and 
this  given  number  of  its  teeth. 


206.    To  -determine  the  pitch  of  the  teeth  of  a  wheel,  knowing 
the  work  to  be  transmitted  by  the  wheel. 

Let  U  represent  the  number  of  units  of  work  to  be  trans- 
mitted by  the  wheel  per  minute,  m  the  number  of  revolutions 
to  be  made  by  it  per  minute,  n  the  number  of  the  teeth  to 
be  cut  iri  it,  T  the  pitch  of  each  tooth  in  feet,  P  the  pressure 
upon  each  tooth  in  pounds. 

Therefore  nT  represents  the  circumference  of  the  pitch 
circle  of  the  wheel,  and  mnT  represents  the  space  in  feet 
described  by  it  per  minute.  Now  U  represents  •  the  work 

transmitted  by  it  through  this  space  per  minute,  therefore  —  1~ 

represents  the  mean  pressure  under  which  this  work  is  trans- 
mitted (Art.  50.)  ; 


The  pitch  T  of  the  teeth  would  evidently  equal  twice  the 
breadth  of  each  tooth,  if  the  spaces  between  the  teeth  were 
equal  in  width  to  the  teeth.  In  order  that  the  teeth  of 
wheels  which  act  together  may  engage  with  one  another  and 
extricate  themselves,  with  facility,  it  is  however  necessary 
that  the  pitch  should  exceed  twice  the  breadth  of  the  tooth 
by  a  quantity  which  varies  according  to  the  accuracy  of  the 
construction  of  the  wheel  from  TVth  to  TV  th  of  the  breadth.* 

Since  the  pitch  T  of  the  tooth  is  dependant  upon  its 
breadth,  and  that  the  breadth  of  the  tooth  is  dependant,  by 
the  theory  of  the  strength  of  materials,  upon  the  pressure  P 
which  it  sustains,  it  is  evident  that  the  quantity  P  in  the 
above  equation  is  a  function  of  T.  This  functionf  may  be 
assumed  of  the  form 

*  For  a  full  discussion  of  this  subject  see  Professor  Willis's  Principles  of 
Mechanism,  Arts.  107-112. 

f  See  Appendix,  on  the  dimensions  of  wheels. 


244  THE  STRENGTH  OF  TEETH. 


T=c  4/P (233) ; 

where  c  is  a  constant  dependant  for  its  amount  upon  the 
nature  of  the  material  out  of  which  the  tooth  is  formed. 
Eliminating  P  between  this  equation  and  the  last,  and  solving 
in  respect  to  T, 


The  number  of  units  of  work  transmitted  by  any  machine 
per  minute  is  usually  represented  in  horses'  power,  one 
horse's  power  being  estimated  at  33,000  units,  so  that  the 
number  of  horses'  power  transmitted  by  the  machine  means 
the  number  of  times  33,000  units  of  work  are  transmitted  by 
it  every  minute,  or  the  number  of  times  33,000  must  be 
taken  to  equal  the  number  of  units  of  work  transmitted  by 
it  every  minute.  If  therefore  H  represent  the  number  of 
horses'  power  transmitted  by  the  wheel,  then  U=33,OOOH. 
Substituting  this  value  in  the  preceding  equation,  and  repre- 
senting the  constant  33,000c2  by  C8,  we  have 

— '.  .  (234). 

mn 

The  values  of  the  constant  C  for  teeth  of  different  mate- 
rials are  given  in  the  Appendix. 


207.  To  determine  the  radius  of  the  pitch  circle  of  a  wheel 
which  shall  contain  n  teeth  of  a  given  pitch. 

Let  AB  represent  the  pitch  T  of  a  tooth, 
and  let  it  be  supposed  to  coincide  with  its 
chord  AMB.  Let  E  represent  the  radius  AC 
of  the  pitch  circle,  and  n  the  number  of  teeth 
to  be  cut  upon  the  wheel. 

Now  there  are  as  many  pitches  in  the  cir- 
cumference as  teeth,  therefore  the  angle  ACB 

subtended  by  each  pitch  is  represented  by — . 

Also  T=2AM=2AC  sin.  £ACB=2Esin.  -  ; 

n 

""• (235). 


TO   DESCRIBE   EPICYCLOIDAL    TEETH.  245 


208.  To  make  the  pattern  of  an  epicycloidal  tooth. 

Having  determined,  as  above, 
the  pitch  of  the  teeth,  and  the 
radius  of  the  pitch  circle,  strike 
an  arc  of  the  pitch  circle  on  a 
thin  piece  of  oak  board  or  me- 
tal plate,  and,  with  a  fine  saw, 
cut  the  board  through  along 
the  circumference  of  this  cir- 
cle, so  as  to  divide  it  into  two 
parts,  one  having  a  convex  and 
the  other  a  corresponding  con- 
cave circular  edge.  Let  EF 
represent  one  of  these  portions 
of  the  board,  and  GH  another. 

Describe  an  arc  of  the  pitch  circle  upon  a  second  board  or 
plate  from  which  the  pattern  is  to  be  cut.  Let  MN  repre- 
sent this  arc.  Fix  the  piece  GH  upon  this  board,  so  that  its 
circular  edge  may  accurately  coincide  with  the  circumference 
of  the  arc  SOT.  Take,  then,  a  circular  plate  D  of  wood  or 
metal,  of  the  dimensions  which  it  is  proposed  to  give  to  the 
generating  circle  of  the  epicycloid  ;  and  let  a  small  point  of 
steel  P  be  fixed  in  it,  so  that  this  point  may  project  slightly 
from  its  inferior  surface,  and  accurately  coincide  with  its  cir- 
cumference. Having  set  off  the  width  AB  of  the  toothj  so 
that  twice  this  width  increased  by  from  TVth  to  TVth  of  that 
width  (according  to  the  accuracy  of  workmanship  to  be 
attained)  may  equal  the  pitch,  cause  the  circle  D  to  roll  upon 
the  convex  edge  GK  of  the  board  GH,  pressing  it,  at  the 
same  time,  slightly  upon  the  surface  of  the  board  on  which 
the  arc  IVCST  is  described,  and  from  which  the  pattern  is  to  he 
cut,  having  caused  the  steel  point  in  its  circumference  first 
of  all  to  coincide  with  the  point  A ;  an  epicycloidal  arc  AP 
will  thus  be  described  by  the  point  P  upon  the  surface  MN". 
Describe,  similarly,  an  epicycloidal  arc  BE  through  the  point 
B,  and  let  them  meet  in  E. 

Let  the  board  GHnow  be  removed,  and  let  EF  be  applied 
and  fixed,  so  that  its  concave  edge  may  accurately  coincide 
with  the  circular  arc  MK  With  the  same  circular  plate  D 
pressed  upon  the  concave  edge  of  EF,  and  made  to  roll  upon 
it,  cause  the  point  in  its  circumference  to  describe  in  like 
manner,  upon  the  surface  of  the  board  from  which  the  pat- 
tern is  to  be  cut,  a  hypococloidal  arc  BH  passing  through  the 


246 


TO   DESCKIBE   EPICYCLOIDAL   TEETH. 


point  B,  and  another  AI  passing  through  the  point  A.  HEI 
will  then  represent  the  form  of  a  tooth,  which  will  work  cor- 
rectly (Art.  202.)  with  the  teeth  similarly  cut  upon  any  other 
wheel ;  provided  that  the  pitch  of  the  teeth  so  cut  upon  the 
other  wheel  be  equal  to  the  pitch  of  the  teeth  upon  this,  and 
provided  that  the  same  generating  circle  D  l)e  used  to  strike 
the  curves  upon  the  two  wheels. 


209.  To  determine  the  proper  lengths  of  epicycloidal  teeth. 

The  general  forms  of  the  teeth  of  wheels  being  determined 
by  tlie  method  explained  in  the  preceding  article,  it  remains 
to  cut  them  off  of  such  lengths  as  may  cause  them  succes- 
sively to  take  up  the  work  from  one  another,  and  transmit  it 
under  the  circumstances  most  favourable  to  the  economy  of 
its  transmission,  and  to  the  durability  of  the  teeth. 

In  respect  to  the  economy  of  the  power  in  its  transmission, 
it  is  customary,  for  reasons  to  be  assigned  hereafter,  to  pro- 
vide that  no  tooth  of  the  one  wheel  should  come  into  action 
with  a  tooth  of  the  other  until  both  are  in  the  act  of  passing 
through  the  line  of  centres.  This  condition  may  be  satisfied 
in  all  cases  where  the  numbers  of  teeth  on  neither  of  the 
wheels  is  exceedingly  small,  by  properly  adjusting  the 
lengths  of  the  teeth.  Let  two  of  the  teeth  of  the  wheels  be 
in  contact  at  the  point  A  in  the  line  CD,  joining  the  centres 
of  the  two  wheels  ;  and  let  the  wheel  whose  centre  is  C  be 
the  driving  wheel.  Let  AH  be  a  portion  of  the  circumfe- 
rence of  the  generating  circle  of  the  teeth,  then  will  the 
points  A  and  !>,  where  this  circle  intersects  the  edges  of  the 


/\ 

X 

f  1*  \ 

^==:Sitcf^ 

0 

&•«".  -•:_ 

PI* 

H* 


v      \ 


teeth  O  and  K  of  the  driving  wheel,  be  points  of  contact 


TO   DESCRIBE   EPICYCLOIDAL    TEETH.  24:? 

with  the  edges  of  the  teeth  M  and  L  of  the  driven  wheel 
(Art.  202.).  JSTow,  since  each  tooth  is  to  come  into  action 
only  when  it  comes  into  the  line  of  centres,  it  is  clear  that 
the  tooth  L  must  have  been  driven  by  K  from  the  time  when 
their  contact  was  in  the  line  of  centres,  until  they  have  come 
into  the  position  shown  in  the  figure,  when  the  point  of  con- 
tact of  the  anterior  face  of  the  next  tooth  O  of  the  driving 
wheel  with  the  flank*  of  the  next  tooth  M  of  the  driven 
wheel  has  just  passed  into  the  line  of  centres  ;  and  since  the 
tooth  O  is  now  to  take  up  the  task  of  impelling  the  driven 
wheel,  and  the  tooth  K  to  yield  it,  all  that  portion  of  the 
last-mentioned  tooth  which  lies  beyond  the  point  B  may  evi- 
dently be  removed  ;  and  if  it  ~be  thus  removed,  then  the  tooth 
K,  passing  out  of  contact,  will  manifestly,  at  that  period  of 
the  motion,  yield  all  the  driving  strain  to  the  tooth  O,  as  it 
k«  required  to  do.  In  order  to  cut  the  pattern  tooth  of  the 

proper  length,  so  as  to  satisfy 
the  proposed  condition,  we  have 
only  then  to  take  A.a  (see  the 
accompanying  figure)  equal  to 
the  pitch  of  the  tooth,  and  to 
bring  the  convex  circumference 
of  the  generating  circle,  so  as 
to  touch  the  convex  circumfe- 
rence of  the  arc  MK  in  that 
point  a ;  the  point  of  intersec- 
tion e  of  this  circle  with  the 
N8  face  AE  of  the  tooth  will  be 
the  last  acting  point  of  the  tooth  ;  and  if  a  circle  be  struck 
from  the  centre  of  the  pitch  circle  passing  through  that 
point,  all  that  portion  of  the  tooth  which  lies  beyond  this  cir- 
cle may  be  cut  off.f 

The  length  of  the  tooth  on  the  wheel  intended  to  act  with 
this,  may  be  determined  in  like  manner. 

210.  In  the  preceding  article  we  have  supposed  the  same 
generating  circle  to  be  used  in  striking  the  entire  surfaces 
of  the  teeth  on  both  wheels.  It  is  not  however  necessary  to 

*  That  portion  of  the  edge  of  the  tooth  which  is  without  the  pitch  circle  is 
called  its  face,  that  within  it  its  flank. 

f  The  point  e  thus  determined  will,  in  some  cases,  fall  beyond  the  extremity 
E  of  the  tooth.  In  such  cases  it  is  therefore  impossible  to  cut  the  tooth  of 
such  a  length  as  to  satisfy  the  required  conditions,  viz.  that  it  shall  drive  only 
after  it  has  passed  the  line  of  centres.  A  full  discussion  of  these  impossible 
cases  will  be  found  in  Professor  Willis's  work  (Arts.  102-104.). 


24:8 


TO   DESCRIBE   EPICYCLOIDAJL   TEETH. 


the  correct  working  of  the  teeth,  that  the  same  circle  should 
thus  be  used  in  striking  the  entire  surfaces  of  ttwo  teeth 
which  act  together,  but  only  that  the  generating  circle  of 
every  two  portions  of  the  two  teeth  which  come  into  actual 
contact  should  be  the  same.  Thus  the  flank  of  the  driving 
tooth  and  the  face  of  the  driven  tooth  being  in  contact  at 


P  in  the  accompanying  figure,*  this  face  of  the  one  tooth 
and  flank  of  the  other  must  be  respectively  an  epicycloid 
and  a  hypocycloid  struck  with  the  same  generating  circle. 
Again,  the  face  of  a  driving  tooth  and  the  flank  of  a  driven 
tooth  being  in  contact  at  Q,  these,  too,  must  be  struck  by 
the  same  generating  circle.  But  it  is  evidently  unnecessary 
that  the  generating  circle  used  in  the  second  case  should  be 
the  same  as  that  used  in  the  first.  Any  generating  circle 
will  satisfy  the  conditions  in  either  case  (Art.  202.),  provided 
it  be  the  same  for  the  epicycloid  as  for  the  hypocycloid 
which  is  to  act  with  it. 

According  to  a  general  (almost  a  universal)  custom  among 
mechanics,  two  different  generating  circles  are  thus  used  for 
striking  the  teeth  on  two  wheels  which  are  to  act  together, 
the  diameter  of  the  generating  circle  for  striking  the  faces 
of  the  teeth  on  the  one  wheel  being  equal  to  the  radius  of 
the  pitch  circle  of  the  other  wheel.  Thus  if  we  call  the 
wheels  A  and  B,  then  the  epicycloidal  faces  of  the  teeth  on 
A,  and  the  corresponding  hypocycloidal  flanks  on  B,  are 
generated  by  a  circle  whose  diameter  is  equal  to  the  radius 
of  the  pitch  circle  of  B.  The  hypocycloidal  flanks  of  the 
teeth  on  B  thus  become  straight  lines  (Art.  203.),  whose 
directions  are  those  of  radii  of  that  wheel.  In  like  manner, 

*  The  upper  wheel  is  here  supposed  to  drive  the  lower. 


TO  DESCRIBE  EPICYCLOIDAL  TEETH.       e    249 

the  epicycloidal  faces  of  the  teeth  on  B,  and  the  correspond- 
ing hypocycloidal  flanks  of  the  teeth  on  A,  are  struck  by  a 
circle  whose  diameter  is  equal  to  the  radius  of  the  pitch  cir- 
cle of  A ;  so  that  the  hypocycloidal  flanks  of  the  teeth  of  A 
become  in  like  manner  straight  lines,  whose  directions  are 
those  of  radii  of  the  wheel  A.  By  this  expedient  of  using 
two  different  generating  circles,  the  flanks  of  the  teeth  on 
both  wheels  become  straight  lines,  and  the  faces  only  are 
curved.  The  teeth  shown  in  the  above  figure  are  of  this 
form.  The  motive  for  giving  this  particular  value  to  the 
generating  circle  appears  to  be  no  other  than  that  saving  of 
trouble  which  is  offered  by  the  substitution  of  a  straight  for 
a  curved  flank  of  the  tooth.  A  more  careful  consideration 
of  the  subject,  however,  shows  that  there  is  no  real  economy 
of  labour  in  this.  In  the  first  place,  it  renders  necessary 
the  use  of  two  different  generating  circles  or  templets  for 
striking  the  teeth  of  any  given  wheel  or  pinion,  the  curved 
portions  of  the  teeth  of  the  wheel  being  struck  with  a  circle 
whose  diameter  equals  half  the  diameter  of  the  pinion,  and 
the  curved  portions  of  the  teeth  of  the  pinion  with  a  circle 
whose  diameter  equals  half  that  of  the  wheel.  Now,  one 
generating  circle  would  have  done  for  both,  had  the  work- 
man been  contented  to  make  the  flanks  of  his  teeth  of  the 
hypocycloidal  forms  corresponding  to  it.  But  there  is  yet  a 
greater  practical  inconvenience  in  this. method.  A  wheel 
and  pinion  thus  constructed  will  only  work  with  one  another  / 
neither  will  work  truly  any  third  wheel  or  pinion  of  a  differ- 
ent number  of  teeth,  although  it  have  the  same  pitch.  Thus 
the  wheels  A  and  B  having  each  a  given  number  of  teeth, 
and  being  made  to  work  with  one  another,  will  neither  of 
them  work  truly  with  C  of  a  different  number  of  teeth  of 
the  same  pitch.  For  that  A  may  work  truly  with  C,  the 
face  of  its  teeth  must  be  struck  with  a  generating  circle, 
whose  diameter  is  half  that  of  C :  but  they  are  struck  with 
a  circle  whose  diameter  is  half  that  of  B ;  the  condition  of 
uniform  action  is  not  therefore  satisfied.  Now  let  us  sup- 
pose that  the  epicycloidal  faces,  and  the  hypocycloidal  flanks 
of  all  the  teeth  A,  B,  and  C  had  been  struck  with  the  same 
generating  circle,  and  that  all  three  had  been  of  the  same 
pitch,  it  is  clear  that  any  one  of  them  would  then  have 
worked  truly  with  any  other,  and  that  this  would  have  been 
equally  true  of  any  number  of  teeth  of  the  same  pitch. 
Thus,  then,  the  machinist  may,  by  the  use  of  the  same  gen- 
erating circle,  for  all  his  pattern  wheels  of  the  same  pitch,  so 
construct  them,  as  that  any  one  wheel  of  that  pitch  shall 


250  TO   DESCRIBE   EPICYCLOIDAL   TEETH. 

work  with  any  other.  This  offers,  under  many  circumstances 
great  advantages,  especially  in  the  very  great  reduction  of 
the  number  of  patterns  which  he  will  be  required  to  keep. 
There  are,  moreover,  many  cases  in  which  some  arrange- 
ment similar  to  this  is  indispensable  to  the  true  working  of 
the  wheels,  as  when  one  wheel  is  required  (which  is  often 
the  case)  to  work  with  two  or  three  others,  of  different  num- 
bers of  teeth,  A  for  instance  to  turn  B  and  C ;  by  the  ordi- 
nary method  of  construction  this  combination  would  be 
impracticable,  so  that  the  wheels  should  work  truly.  Any 
generating  circle  common  to  a  whole  set  of  the  same  pitch, 
satisfying  the  above  condition,  it  may  be  asked  whether 
there  is  any  other  consideration  determining  the  best  dimen- 
sions of  this  circle.  There  is  such  a  consideration  arising 
out  of  a  limitation  of  the  dimensions  of  the  generating  circle 
of  the  hypocycloidal  portion  of  the  tooth  to  a  diameter  not 
greater  than  half  that  of  its  base.  As  long  as  it  remains 
within  these  limits,  the  hypocycloidal  generated  by  it  is  of 
that  concave  form  by  which  the  flank  of  the  tooth  is  made 
to  spread  itself,  and  the  base  of  the  tooth  to  widen ;  when 
it  exceeds  these  limits,  the  flank  of  the  tooth  takes  the  con- 
vex form,  the  base  of  the  tooth  is  thus  contracted,  and  its 
strength  diminished.  Since  then,  the  generating  circle 
should  not  have  a  diameter  greater  than  half  that  of  any  of 
the  wheels  of  the  set  for  which  it  is  used,  it  will  manifestly 
be  the  greatest  which  will  satisfy  this  condition  when  its 
diameter  is  equal  to  half  that  of  the  least  wheel  of  the  set. 
Now  no  pinion  should  have  less  than  twelve  or  fourteen 
teeth.  Half  the  diameter  of  a  wheel  of  the  proposed  pitch, 
which  has  twelve  or  fourteen  teeth,  is  then  the  true  diame- 
ter or  the  generating  circle  of  the  set.  The  above  sugges- 
tions are  due  to  Professor  Willis.* 


*  Professor  Willis  has  suggested  a  new  and  very  ingenious  method  of 
striking  the  teeth  of  wheels  by  means  of  circular  arcs.  A  detailed  description 
of  this  method  has  been  given  by  him  in  the  Transactions  of  the  Institution 
of  Civil  Engineers,  vol.  ii.,  accompanied  by  tables,  &c.,  which  render  its  prac« 
tical  application  exceedingly  simple  and  easy. 


TO   DESCEIBE   INVOLUTE   TEETH. 


251 


211.   To   DESCRIBE   INVOLUTE   TEETH. 

Let  AD  and  AG  represent  the  pitch  circles  of 
two  wheels  intended  to  work  together.  Draw  a 
straight  line  FE  through  the  point  of  contact  A 
of  the  pitch  circles  and  inclined  to  the  line  of 
centres  CAB  of  these  wheels  at  a  certain  angle 
FAG,  the  influence  of  the  dimensions  of  which 
on  the  action  of  the  teeth  will  hereafter  be  ex- 
plained, but  which  appears  usually  to  be  taken 
not  less  than  80°.*  Describe  two  circles  eEK 
and  /*FL  from  the  centres  B  and  C,  each  touching  the 
straight  line  EF.  These  circles  are  to  be  taken  as  the  bases 
from  which  the  involute  faces  of  the  teeth  are  to  be  struck. 
It  is  evident  (by  the  similar  triangles  ACF  and  AEB)  that 
their  radii  CF  and  BE  will  be  to  one  another  as  the  radii 
CA  and  BA  of  the  pitch  circles,  so  that  the  condition  neces- 
sary (Art.  201.)  to  the  correct  action  of  the  teeth  of  the 
wheels  will  be  satisfied,  provided  their  faces  be  involutes  to 


these  two  circles.  Let  AG  and  AH  in  the  above  figure 
represent  arcs  of  the  pitch  circles  of  the  wheels  on  an 
enlarged  scale,  and  0E,  /X,  corresponding  portions  of  the 
circles  eEK  and  yFL  of  the  preceding  figure.  Also  let  A.a 
represent  the  pitch  of  one  of  the  teeth  of  either  wheel. 
Through  the  points  A  and  a  describe  involutes  ef  and  mn.\ 

*  See  Camus  on  the  Teeth  of  Wheels,  by  Hawkins,  p.  168. 

f  Mr.  Hawkins  recommends  the  following  as  a  convenient  method  of  striking 
involute  teeth,  in  his  edition  of  "  Camus  on  the  Teeth  of  Wheels,"  p.  166.  Take 
a  thin  board,  or  a  plate  of  metal,  and  reduce  its  edge  MX  so  as  accurately  to 


252  TO   DESCEIBE   INVOLUTE  TEETH. 

Let  5  be  the  point  where  the  line  EF  intersects  the  involute 
mn ;  then  if  the  teeth  on  the  two  wheels  are  to  be  nearly  of 
the  same  thickness  at  their  bases,  bisect  the  line  AJb  in  c ;  or 
if  they  are  to  be  of  different  thicknesses,  divide  the  line  Ab 
in  c  in  the  same  proportion*,  and  strike  through  the  point  c 
an  involute  curve  hg,  similar  to  ef,  but  inclined  in  the  oppo- 
site direction.  If  the  extremity^  of  the  tooth  be  then  cut 
off  so  that  it  may  just  clear  the  circumference  of  the  circle 
y*L,  the  true  form  of  the  pattern  involute  tooth  will  be 
obtained/ 

There  are  two  remarkable  properties  of  involute  teeth,  by 
the  combination  of  which  they  are  distinguished  from  teeth 
of  all  other  forms,  and  cceteris  paribus  rendered  greatly  pre- 
ferable to  all  others.  The  lirst  of  these  is,  that  any  two 
wheels  having  teeth  of  the  involute  form,  and  of  the  same 
pitch,t  will  work  correctly  together,  since  the  forms  of  the 
teeth  on  any  one  sucli  wheel  are  entirely  independent  of 
those  on  the  wheel  which  is  destined  to  work  with  it  (Art. 
201.)  Any  two  wheels  with  involute  teeth  so  made  to  work 
together  will  revolve  precisely  as  they  would  by  the  actual 
contact  of  two  circles,  whose  radii  may  be  found  by  divid- 
ing the  line  joining  their  centres  in  the  proportion  of  the 
radii  of  the  generating  circles  of  the  involutes.  This  pro- 
perty involute  teeth  possess,  however,  in  common  with  the 
epicycloidal  teeth  of  different  wheels,  all  of  which  are  struck 
with  the  same  generating  circle  (Art.  210.)  The  second  no 
less  important  property  of  involute  teeth — a  property  which 
distinguishes  them  from  teeth  of  all  other  forms — is  this, 
that  they  work  equally  well,  however  far  the  centres  of  the 


coincide  with  the  circular  arc 
-JB  eE,  and  let  a  piece  of  thin 
watch-spring  OR,  having  two 
projecting  points  upon  it  as 
shown  at  P,  and  which  is  of  a 
width  equal  to  the  thickness  of  the  plate,  be  fixed  upon  its  edge  by  means  of 
a  screw  0.  Let  the  edge  of  the  plate  be  then  made  to  coincide  with  the  arc 
eE  in  such  a  position  that,  when  the  spring  is  stretched,  the  point  P  in  it  may 
coincide  with  the  point  from  which  the  tooth  is  to  be  struck ;  and  the  spring 
being  kept  continually  stretched,  and  wound  or  unwound  from  the  circle,  the 
involute  arc  is  thus  to  be  described  by  the  point  P  upon  the  face  of  the  board 
from  which  the  pattern  is  to  be  cut. 

*  This  rule  is  given  by  Mr.  Hawkins  (p.  170.);  it  can  only  be  an  approxima- 
tion, but  may  be  sufficiently  near  to  the  truth  for  practical  purposes.  It  is  to 
be  observed  that  the  teeth  may  have  their  bases  in  any  other  circles  than 
those,  /L  and  eE,  from  which  the  involutes  are  struck. 

f  The  teeth  being  also  of  equal  thicknesses  at  their  bases,  the  method  of 
ensuring  which  condition  has  been  explained  above. 


THE   TEETH   OF   A   BACK   AND   PINION.  253 

wheels  are  removed  asunder  from  one  another  /  so  that  the 
action  of  the  teeth  of  two  wheels  is  not  impaired  when 
their  axes  are  displaced  by  that  wearing  of  their  brasses  or 
collars,  which  soon  results  from  a  con- 
tinued and  a  considerable  strain.  The 
existence  of  this  property  will  readily  be 
admitted,  if  we  conceive  AG  and  BH  to 
represent  the  generating  circles  o/bases 
of  the  teeth,  and  these  to  be  placed  with 
their  centres  Cj  and  C2  any  distance 
asunder,  a  band  AB  (p.  235.,  note)  passing 
round  both,  and  a  point  T  in  this  band 
generating  a  curve  mn,  m'  n'  on  the  plane 
of  each  of  the  circles  as  they  are  made  to 
revolve  under  it.  It  has  been  shown  that 
these  curves  mn  and  mr  n'  will  represent  the  faces  of  two 
teeth  which  will  work  truly  with  one  another ;  moreover, 
that  these  curves  are  respectively  involutes  of  the  two 
circles  AG  and  BH,  and  are  therefore  wholly  independent 
in  respect  to  their  forms  of  the  distances  of  the  centres  of 
the  circles  from  one  another,  depending  only  on  the  dimen- 
sions of  the  circles.  Since  then  the  circles  would  drive  at 
any  distance  correctly  by  means  of  the  band ;  since,  more- 
over, at  every  such  distance  they  would  be  driven  by  the 
curves  mn  and  m'n'  precisely  as  by  the  band ;  and  since 
these  curves  would  in  every  such  position  be  the  same 
curves,  viz.  involutes  of  the  two  circles,  it  follows  that  the 
same  involute  curves  inn  and  m'n'  would  drive  the  circles 
correctly  at  whatever  distances  their  centres  were  placed ; 
and,  therefore,  that  involute  teeth  would  drive  these  wheels 
correctly  at  whatever  distances  the  axes  of  those  wheels 
were  placed. 


THE  TEETH  or  A  KAOK  AND  PINION. 

212.  To  determine  the  pitch  circle  of  the  pinion.  Let  H 
represent  the  distance  through  which  the  rack  is  to  be 
moved  by  each  tooth  of  the  pinion,  and  let  these  teeth  be 
N  in  number ;  then  will  the  rack  be  moved  through  the 
space  N  .  H  during  one  complete  revolution  of  the  wheel. 
!N"ow  the  rack  and  pinion  are  to  be  driven  by  the  action  of 
their  teeth,  as  they  would  by  the  contact  of  the  circum- 


254 


THE    TEETH    OF    A    HACK    AND    PTNIOX. 


ference  of  the  pitch  circle  of  the 
pinion  with  the  plane  face  of  the 
rack,  so  that  the  space  moved  through 
by  the  rack  during  one  complete 
revolution  of  the  pinion  must  pre- 
cisely equal  the  circumference  of  the 
pitch  circle  of  the  pinion.  If,  there- 
fore we  call  R  the  radius  of  the 
pitch  circle  of  the  pinion,  then 


213.  To  describe  the  teeth  of  the 
pinion,  those  of  the  rack  being 
straight.  The  properties  which  have 
been  shown  to  belong  to  involute 
teeth  (Art.  201.)  manifestly  obtain, 
however  great  may  be  the  dimensions  of  the  pitch  circle 
of  their  wheels,  or  whatever  disproportion 
may  exist  between  them.  Of  two  wheels 
OF  and  OE  with  involute  teeth  which 
work  together,  let  then  the  radius  of  the 
pitch  circle  of  one  OF  become  infinite,  its 
circumference  will  then  become  a  straight 
line  represented  by  the  face  of  a  rack. 
Whilst  the  radius  C2O  of  the  pitch  circle 
OF  thus  becomes  infinite,  that  C2B  of  the 
circle  from  which  its  involute  teeth  are 
struck  (bearing  a  constant  ratio  to  the  first) 
will  also  become  infinite,  so  that  the  invo- 
lute m'n'  will  become  a  straight  line*  perpendicular  to  the 
line  AB  given  in  position.  The  involute  teeth  on  the 
wheel  OF  will  thus  become  straight  teeth  (see  fig.  1.),  hav- 
ing their  faces  perpendicular  to  the  line  AB  determined  by 
drawing  through  the  point  O  a  tangent  to  the  circle  AC, 
from  which  the  involute  teeth  of  the  pinion  are  struck.  If 
the  circle  AC  from  which  the  involute  teeth  of  the  pinion 
are  struck  coincide  with  its  pitch  circle,  the  line  AB  becomes 

*  For  it  is  evident  that  the  extremity  of  a  line  of  infinite  length  unwinding 
itself  from  the  circumference  of  a  circle  of  infinite  diameter  will  describe, 
through  a  finite  space,  a  straight  line  perpendicular  to  the  circumference  of 
the  circle.  The  idea  of  giving  an  oblique  position  to  the  straight  faces  of  the 
teeth  of  a  rack  appears  first  to  have  occurred  to  Professor  Willis. 


THE   TEETH   OF   A   RACK   AND   PINION. 


255 


parallel  to  the  face  of  the  rack,  and  the  edges  of  the  teeth 
of  the  rack  perpendicular  to  its  face  (fig.  2.). 

Now, -the  involute  teeth  of  the  one  wheel  have  remained 
unaltered,  and  the  truth  of  their  action  with  teeth  of  the 
other  wheel  .has  not  been  influenced  by  that  change  in  the 
dimensions  of  the  pitch  circle  of  the  last,  which  has  con- 
verted it  into  a  rack,  and  its  curved  into  straight  teeth. 
Thus,  then,  it  follows,  that  straight  teeth  upon  a  rack,  work 
truly  with  involute  teeth  upon  a  pinion.  Indeed  it  is  evi- 


ct-) 


(2.) 


dent,  that  if  from  the  point  of  contact  P  (fig.  2.)  of  such  an 
involute  tooth  of  the  pinion  with  the  straight  tooth  of  a 
rack  we  draw  a  straight  line  *PQ  parallel  to  the  face  ab  of 
the  rack,  that  straight  line  will  be  perpendicular  to  the 
surfaces  of  both  the  teeth  at  their  point  of  contact  P,  and 
that  being  perpendicular  to  the  face  of  the  involute  tooth, 
it  "will  also  touch  the  circle  of  which  this  tooth  is  the  invo- 
lute in  the  point  A,  at  which  the  face  ab  of  the  rack  would 
touch  that  circle  if  they  revolved  by  mutual  contact.  Thus, 
then,  the  condition  shown  in  Art.  199.  to  be  necessary  and 
sufficient  to  the  correct  action  of  the  teeth,  namely,  that  a 
line  drawn  from  their  point  of  contact,  at  any  time,  to  the 
point  of  contact  of  their  pitch  circles,  is  satisfied  in  respect 
to  these  teeth.  Divide,  then,  the  circumference  of  the 
pitch  circle,  determined  as  above  (Art.  212.),  into  N  equal 


256 


THE   TEETH   OF   A   BACK   AND    PINION. 


parts,  and  describe  (Art.  211.)  a  pattern  involute  tooth  from 
the  circumference  of  the  pitch  circle,  limiting  the  length  of 
the  face  of  the  tooth  to  a  little  more  than  the  length  5P  of 
the  involute  curve  generated  by  unwinding  a  length  AP  of 
the  flexible  line  equal  to  the  distance  H  through  which  the 
rack  is  to  be  moved  by  each  tooth  of  the  pinion.  The 
straight  teeth  of  the  rack  are  to  be  cut  of  the  same  length, 
and  the  circumference  of  the  pitch  circle  and  the  face  ao  of 
the  rack  placed  apart  from  one  another  by  a  little  more 
than  this  length. 

It  is  an  objection  to  this  last  application  of  the  involute 
form  of  tooth  for  a  pinion  working  with  a  rack,  that  the 
point  P  of  the  straight  tooth  of  the  rack  upon  which  it  acts 
is  always  the  same,  being  determined  by  its  intersection  with 
a  line  AP  touching  the  pitch  circle,  and  parallel  to  the  face 
of  the  rack.  The  objection  does  not  apply  to  the  preceding, 
the  case  (fig.  1.)  in  which  the  straight  faces  of  each  tooth  of 
the  rack  are  inclined  to  one  another.  By  the  continual 
action  upon  a  single  point  of  the  tooth  of  the  rack,  it  is 
liable  to  an  excessive  wearing  away  of  its  surface. 


214.  To  describe  the  teeth  of  the  pinion,  the  teeth  of  the  rack 
leing  curved. 

This  may  be  done  by  giving  to  the  face  of  the  tooth  of 


the  rack  a  cycloidal  form,  and  making  the  face  of  the  tooth 
of  the  pinion  an  epicycloid,  as  will  be  apparent  if  we  con- 
K.  ceive  the  diameter  of  the  circle  whose 

centre  is  C  (see  fig.  p.  236.)  to  become 
infinite,  the  other  two  circles  remain- 
ing unaltered.  Any  finite  portion  of 
the  circumference  of  this  infinite  circle 
will  then  become  a  straight  line.  Let 
AE  in  the  accompanying  figure  repre- 


THE   TEETH   OF   A   WHEEL   WITH   A   LANTERN.  257 

sent  such  a  portion,  and  let  PQ  and  PR  represent,  as 
before,  curves  generated  by  a  point  P  in  the  circle  whose 
centre  is  D,  when  all  three  circles  revolve  by  their  mutual 
contact  at  A.  Then  are  PR  and  PQ  the,  true  forms  of  the 
teeth  which  would  drive  the  circles  as  they  are  driven  by 
their  mutual  contact  at  A  (Art.  202).  Moreover,  the  curve 
PQ  is  the  same  (Art.  199.)  as  would  be  generated  by  the 
point  P  in  the  circumference  of  APH ;  if  that  circle  rolled 
upon  the  circumference  AQF,  it  is  therefore  an  epicycloid  / 
and  the  curve  PR  is  the  same  as  would  be  generated  by  the 
point  P,  if  the  circle  APH  rolled  upon  the  circumference 
or  straight  line  AE,  it  is  therefore  a  cycloid.  Thus  then  it 
appears,  that  after  the  teeth  have  passed  the  line  of  centres, 
when  the  face  of  the  tooth  of  the  pinion  is  driving  the  flank 
of  the  tooth  of  the  rack,  the  former  must  have  an  epicy- 
cloidal,  and  the  latter  a  cycloidal  form.  In  like  manner,  by 
transferring  the  circle  APH  to  the  opposite  side  of  AE,  it 
may  be  shown,  that  before  the  teeth  have  passed  the  line  of 
centres  when  the  flank  of  the  tooth  of  the  pinion  is  driving 
the  face  of  the  tooth  of  the  wheel,  the  former  must  have  a 
hypocycloidal,  and  the  latter  a  cycloidal  form,  the  cycloid 
having  its  curvature  in  opposite  directions  on  the  flank  and 
the  face  of  the  tooth.  The  generating  circle  will  be  of  the 
most  convenient  dimensions  for  the  description  of  the  teeth 
when  its  diameter  equals  the  radius  of  the  pitch  circle  of 
the  pinion.  The  hypocycloidal  flank  of  the  tooth  of  the 
pinion  will  then  pass  into  a  straight  flank.  The  radius  of 
the  pitch  circle  of  the  pinion  is  determined  as  in  Art.  212., 
and  the  method  of  describing  its  teeth  is  explained  in 
Art.  208. 


£15.    THE   TEETH   OF   A   WHEEL    WORKING    WITH   A   LANTERN   OR 

TRUNDLE. 

In  some  descriptions  of  mill  work  the  ordinary  form  of 
the  toothed  wheel  is  replaced  by  a  contrivance  called  a  lan- 
tern or  trundle,  formed  by  two  circular  discs,  which  are  con- 
nected with  one  another  by  cylindrical  columns  called 
staves,  engaging,  like  the  teeth  of  a  pinion,  with  the  teeth 
of  a  wheel  which  the  lantern  is  intended  to  drive.  This 
combination  is  shown  in  the  following  figure. 

It  is  evident  that  the  teeth  on  the  wheel  which  works  with 
the  lantern  have  their  shape  determined  by  the  cylindrical 


258 


THE   TEETH   OF   A   WHEEL    WITH   A   LANTEBN. 


shape  of  the  staves.     Their  forms  may  readily  be  found  by 
the  method  explained  in  Art.  200. 

Having  determined  npon  the  dimensions  of  the  staves  in 
reference  to  the  strain  they  are  to  be  subjected  to,  and  upon 
the  diameters  of  the  pitch  circles  of  the  lantern  and  wheel, 
and  also  upon  the  pitch  of  the  teeth ;  strike  arcs  AB  and 
AC  of  these  circles,  and  set  off  upon  them 
the  pitches  A.a  and  A5  from  the  point  of 
contact  A  of  the  pitch  circles  (if  the  teeth 
are  first  to  come  into  contact  in  the  line 
of  centres,  if  not,  set  them  off  from  the 
points  behind  the  line  of  centres  where 
the  teeth  are  first  to  come  into  contact). 
Describe  a  circle  #<?,  having  its  centre  in 
AB,  passing  through  #,  and  having  its 
diameter  equal  to  that  of  the  stave,  and  divide  each  of  the 
pitches  Aa  and  A5  into  the  same  number  of  equal  parts 
(say  three).  From  the  points  of  division  A,  a,  /3  in  the 
pitch  A#,  measure  the  shortest  distances  to  the  circle  #0,  and 
with  these  shortest  distances,  respectively,  describe  from  the 
points  of  division  7,  <$  of  the  pitch  A5,  circular  arcs  inter- 
secting one  another ;  a  curve  ah  touching  all  these  circular 
arcs  will  give  the  true  face  of  the  tooth  (Art.  200.).  The 
opposite  face  of  the  tooth  must  be  struck  from  similar  cen- 
tres, and  the  base  of  the  tooth  must  be  cut  so  far  within  the 
pitch  circle  as  to  admit  one  half  of  the  stave  ae  when  that 
stave  passf*s  the  line  of  centres. 


PRESSURES   UPON   WHEELS. 


259 


216.  THE   RELATION    BETWEEN    TWO    PRESSURES  Pt  AND  Pa 

APPLIED   TO   TWO    TOOTHED   WHEELS    IN    THE     STATE   BORDER- 
ING-   UPON   MOTION   BY   THE   PREPONDERANCE   OF   Pa. 

Let  the  influence  of  the  weights  of  the  wheels  be  in  the 
first  place  neglected.  Let  B  and  C  represent  the  centres  of 
the  pitch  circles  of  the  wheels,  A  their  point  of  contact,  P 
the  point  of  contact  of  the  driving  and  driven  teeth  at  any 
period  of  the  motion,  KP  the  direction  of  the  whole 
resultant  pressure  upon  the  teeth  at  their  point  of  contact, 
which  resultant  pressure  is  equal  and  opposite  to  the  resist- 
ance B-  of  the  follower  to  the  driver,  BM  and  CN  perpen- 
diculars from  the  centres  of  the  axes  of  the  wheels  upon  IIP  ; 
and  BD  and  CE  upon  the  directions  of  Pl  and  Pa. 


pi5  p2=radii  of  axes  of  wheels. 

<pl5  <p2=r  limiting  angles  of  resistance  between  the  axes  of 
the  wheels  and  their  bearings. 

Then,  since  Pa  and  B,  applied  to  the  wheel  whose  centre  is 


260  RELATION  OF  THE   DRIVING   AND   WORKING 

B  are  in  the  state  bordering  upon  motion  by  the  preponder- 
ance of  Pj,  and  since  at  and  m^  are  the  perpendiculars  on 
the  directions  of  these  pressures  respectively,  we  have  (equa- 
tion 158) 


where  Lx  represents  the  length  of  the  line  DM  joining  the 
feet  of  the  perpendiculars  BM  and  BD. 

Again,  since  B.  and  P2,  applied  to  the  wheel  whose  centre 
is  C,  are  in  the  state  bordering  upon  motion  by  the  yielding 


of  P3  (Art.  164.), 


where  L2  represents  the  distance  NE  between  the  feet  of  the 
perpendiculars  CE  and  GN.  Eliminating  B.  between  these 
equations,  we  have 


F.=  (^H     '       .?         '    'k (238). 


Now  let  it  be  observed,  that  the  line  AP,  drawn  from  the 
point  of  contact  A  of  the  pitch  circles  to  the  point  of  contact 
P  of  the  teeth  is  perpendicular  to  their  surfaces  at  that  point 
P,  whatever  may  be  the  forms  of  the  teeth,  provided  that 
they  act  truly  with  one  another  (Art.  199.) ;  moreover,  that 
when  the  point  of  contact  P  has  passed  the  line  of  centres, 
as  shown  in  the  figure,  that  point  is  in  the  act  of  moving  on 
the  driven  surface  Ppfrom  the  centre  0,  or  from  P  towards 
p,  so  that  the  friction  of  that  surface  is  exerted  in  the  opposite 
direction,  or  from  p  towards  P ;  whence  it  follows  that  the 
resultant  of  this  friction,  and  the  perpendicular  resistance  aP 
of  the  driven  tooth  upon  the  driver,  lias  its  direction  rP 
within  the  angle  aPp  and  that  it  is  inclined  (Art.  141.)  to  the 
perpendicular  aP  at  an  angle  aPr  equal  to  the  limiting  angle 
of  resistance.  Now  this  resistance  is  evidently  equal  and 
opposite  to  the  resultant  pressure  upon  the  surfaces  of  the 
teeth  in  the  state  bordering  upon  motion  ;  whence  it  follows 
that  the  angle  EPA  is  equal  to  the  limiting  angle  of  resist- 
ance between  the  surfaces  of  contact  of  the  teeth.  Let  this 
angle  be  represented  by  9,  and  let  AP=X.  Also  let  the 


PRESSURES   UPON    WHEELS. 


261 


inclination  PAC  of  AP  to  the  line  of  centres  BC  be  repre- 
sented by  0.  Through  A  draw  An  perpendicular  to  KP,  and 
sAt  parallel  to  it.  Then, 

m1=EM=Kt  +  m==Et+An=BA  sin.  BArf+ AP  sin.  APE- 
Also  BAz5=BOE=:PAC+ APE=d+9 ; 

/.  mv—r^  sin.  (d  +  9)-f  X  sin.  9 (239); 

m,=CN=Cs-s^=Cs-An=CA  sin.  CAs-AP  sin.  APE. 
But  As  is  parallel  to  PE,  therefore  CAs=BOE=0+9; 

/.  ma=7*asin.  (^4-9)— X  sin.  9 (240.). 

Substituting  these  values  of  rax  and  raa  in  the  preceding 
equation, 


-te\H 

l~\aj 


.  9+ 


a, 


—  Xsin.9—      _?sin.93 


217.  In  the  preceding  investigation  the  point  of  contact  P 


262  RELATION   OF   THE   DRIVING   AND   WORKING 

of  the  teeth  of  the  driving  and  driven  wheels  is  supposed  to 
have  passed  the  line  of  centres,  or  to  be  behind  that  line ; 
let  us  now  suppose  it  not  to  have  passed  the  line  of  centres, 
or  to  be  before  that  line. 

It  is  evident  that  in  this  case  the  point  of  contact  P  is  ir 
the  act  of  moving  upon  the  surface  pPq  of  the  driven  tooth 
towards  the  centre  C,  or  from  P  towards  £,  as  in  the  other 
case  it  is  from  the  centre,  or  from  P  towards^.  In  this  case, 
therefore,  the  friction  of  the  driven  surface  is  exerted  in  the 
direction  qP  ;  whence  it  follows,  that  in  this  state  bordering 
upon  motion  the  direction  of  the  resistance  R  of  the  driven 
upon  the  driving  tooth  must  lie  on  the  other  side  of  the 
normal  APQ,  being  inclined  to  it  at  an  angle  APN  equal  to 
the  limiting  angle  of  resistance.  Thus  the  inclination  of  B, 
to  the  normal  APQ  is  in  both  cases  the  same,  but  its  position 
in  respect  to  that  line  is  in  the  one  case  the  reverse  of  its 
position  in  the  other  case.* 

The  same  construction  being  made  as  before, 

m1=BM=B£+£M;=:B£  +  A^=BA.  sin.  BA£+ AP.  sin.  APO. 
Also  BA£=BOK:=BAP— APO=4— 9  ;f 
/.  m1=r1  sin.  (6— <?)+*•  sin.  <p, 

m,=C'N=Cs-sN=Cs— A.n=CA.  sin.  CAs— AP.  sin.  APO. 
But  As  is  parallel  to  PN, 


/.  ma=r2  sin.  (6— 9)— X  sin.  9. 
Substituting  these  values  of  m1  and  ra2  in  equation  (238), 

^sin.  (d— 9)+Xsin.  9  +  I— ijsin.  <p4 
p  _  /a,\   I  *  «.  ; 

x~  fe/ 


ra sin.  (d— 9)— Xsin.  9—  i^-^)sin.  92 

\     ^2     ' 

This  expression  differs  from  the  preceding  (equation  241) 
ily  in  the  substitution  of  (d — 9)  foi 
of  the  numerator  and  denominator. 


only  in  the  substitution  of  (d—  9)  for  (d-f?)  in  the  first  terms 
th 


*  Hence  it  follows,  that  when  the  point  of  contact  is  in  the  act  of  crossing 
the  line  of  centres,  the  direction  of  the  resultant  pressure  R  is  passing  from 
one  side  to  the  other  of  the  perpendicular  APQ  ;  and  therefore  that  when  the 
point  of  contact  is  in  the  line  of  centres,  the  resultant  pressure  is  perpendicu- 
lar to  that  line,  and  the  angle  BOR  a  right  angle  ;  a  condition  which  cannot 
however  be  assumed  to  obtain  approximately  in  respect  to  positions  of  any 
point  of  contact  exceedingly  near  to  the  line  of  centres. 

f  The  angle  6  being  here  taken  as  before  to  represent  the  inclination  BAP 
of  the  line  AP,  joining  the  point  of  contact  of  the  pitch  circles  with  the  point 
of  contact  of  the  teeth,  to  the  line  of  centres. 


PRESSURES   UPON    WHEELS. 


263 


Dividing  numerator  and  denominator  of  the  fraction  in 
the  second  member  of  that  equation  by  sin.  (0  +  <p),  and 
throwing  out  the  factors  rl  and  ra,  we  have 


X  sin.  9  +    J— - 


Xsin.9+(^)sm.  ?2 

1 *  a*  ' 

r2  sin.  (0  +  <p) 


sin. 

Now  it  is  evident,  that  if  in  this  fractional  expression  &— 9 
be  substituted  for  $+<p  the  numerator  will  be  increased  and 
the  denominator  diminished,  so  that  the  value  of  Pj  corre- 
sponding to  any  given  value  of  P2  will  be  increased.  Whence 
it  follows,  that  the  resistance  to  the  motion  of  the  wheels  by 
the  friction  of  the  common  surfaces  of  contact  of  their  teeth 
and  of  the  bearings  of  their  axes  is  greater  when  the  contact 
of  their  teeth  takes  place  before  than  when  it  takes  place, 
at  an  equal  angular  distance,  behind  the  line  of  centres — a 
principle  confirmed  by  the  experience  of  all  practical  me- 
chanists. 


218.  To  DETERMINE  THE  RELATION  OF  THE  STATE  BORDERING 
UPON  MOTION  BETWEEN  THE  PRESSURE  Pt  APPLIED  TO  THE 
DRIVING  WHEEL  AND  THE  RESISTANCE  P2  OPPOSED  TO  THE 


MOTION     OF     THE     DRIVEN     WHEEL,     THE     WEIGHTS 
WHEELS   BEING   TAKEN    INTO   THE   ACCOUNT. 


OF     THE 


Now  let  the  influence  of  the  weights  Wl  and  W2  of  the 
two  wheels  be  taken  into  the  account.  The  pressures  applied 
to  each  wheel  being  now  three  in  number  instead  of  two,  the 
relations  between  Px  and  R,  and  P3  and  K  are  determined 
by  equation  (163)  instead  of  equation  (158).  Substituting 
"W,  and  W2  for  P3  in  the  two  cases,  we  obtain,  instead  of 
equations  (236)  and  (237),  the  following, 


-...(243); 


in  which  equations  M,  and  M,  represent  certain  functions 


264:  RELATION   OF   THE   DRIVING   AND   WORKING 


\ 


determined  (Art.  166.)  by  the  inclinations  of  the  pressures 
Px  and  P2  to  the  vertical. 

Eliminating  B,  between  the  above  equations,  neglecting 
terms  above  the  first  dimensions  in  sin.  9,  and  sin.  92,  and 
multiplying  by  a^ 


,-^sin.  9.  }  -PA  I  mt  +  ~i  sin.  ?,  J  = 

#2  I  d>! 

M3W3 

"Xpi  sm.  9J  +  V  —  mJ*  sm-  ^ 


Substituting  the  values  of  ml  and  wa  from  equations  (239) 
and  (240),  and  neglecting  the  products  of  sin.  9,  sin.  91  and 
sin.  99,  we  obtain 

—  x  sin.  9  --  —  sin.  93  \  — 
n.  9+-      sin.  9,     = 


A     /-..sn. 


sn. 


PRESSURES   UPON   WHEELS.  265 

(245.) 

M 

Now  (Art.  166.)  —  -=ml  cos.  'js  +  flj  cos.  «„,  where  »13  repre- 

®i 

eents  the  inclination  W,FP1  of  "Pl  to  the  vertical,  and  <a3  the 
inclination  ErF  of  E  to  the  vertical.* 

Let  the  inclination  W,BD  of  the  perpendicular  upon  Px  to 
the  vertical  be  represented  by  a1?  that  angle  being  so  mea- 
sured that  the  pressure  Px  may  tend  to  increase  it  ;  let  a,  re- 
present, in  like  manner,  the  inclination  EGG  of  CE  to  the 
vertical;  and  let  (3  represent  the  inclination  ABr  of  the 
line  of  centred  to  the  vertical, 

.-.  .Ii=WlFPl=W1BD-BDF=«1-  J 

i23=E7-F=BOE—  OBr=d+9—  ft  ; 

M 

V  —  l=ml  sin.  aj+flj  cos.  (0+9—  /3). 

a\ 
Similarly  —  ?=raa    cos.   PaGH+«2   cos.  E^.f       Now 


;     and    E^W^rf— ErF,    and 

ErF  was  before  shown  to  be  equal  to  (04-9— (3  • 

M 

V — -=— wa  sin.  aa— 0a  cos.  (4 +  9— p) 

Substituting  the  values  of  m:  and  wa,  from  equations  (239) 
and  (240), 

M 

—±=rl  sin.  (0+9)  sin.  at+X  sm.  a!  sin.  9  + 
a, 

M 

—  —  — /*.,  sin.  (0  +  9)  sin.  aa  +  A.  sin.  aa  sin.  9 — 


a,  cos.  (4+9— 

*  See  note,  p.  172. 

•j-  It  is  to  be  observed  that  the  direction  of  the  arrow  in  the  figure  repre- 
sents that  of  the  resistance  opposed  by  the  driven  wheel  to  the  motion  of  the 
driving  wheel,  so  that  the  direction  of  the  pressure  of  the  driving  upon  the 
driven  wheel  is  opposite  to  that  of  the  arrow. 


266  RELATION   OF   THE   DRIVING    AND    WORKING 

Let  it  be  supposed  that  the  distances  DM  and  EN",  repre- 
sented by  L,  and  L2,  are  of  finite  dimensions,  the  directions 
of  neither  of  the  pressures  P1  and  P2  approaching  to  coinci- 
dence with  the  direction  of  R,  —  a  supposition  which  has  been 
virtually  made  in  deducing  equation  (163)  from  equation 
(161),  on  the  former  of  which  equations,  equations  (243)  de- 
pend. And  let  it  be  observed  that  the  terms  involving  sin.  9 
in  the  above  expressions  (equations  246)  will  be  of  two  di- 
mensions in  <p1?  92  and  9,  when  substituted  in  equation  (245), 
and  may  therefore  be  neglected.  Moreover,  that  in  all  cases 
the  direction  of  HP  is  BO  nearly  perpendicular  to  the  line 
of  centres  BC,  that  in  those  terms  of  equation  (245),  which 
are  multiplied  by  sin.  <px  and  sin.  9,,  the  angle»$  +  9,  or  BOK, 

may  be  asssumed=  -  ;   any  error  which  that  supposition  in- 

2i 

volves,  exceedingly  small  in  itself,  being  rendered  exceed- 
ingly less  by  that  multiplication.  Equations  (246>  will  then 
become 

M,  M, 

—  *=r.  sin.  c^-htfj  sm.  p,  —  -=—  r3  sm.  aa—  az  sm.  p. 

a,  a, 

Substituting  these  values  in  the  first  factor  of  the  second 
member  of-  equation  (245),  and  representing  that  factor  by 
Nr  we  have 


.rjs'j^r^  (/\  sin.  a.-i-a,  sin.  (3)  sin.  9,— 
-L-i 

w 

-T-*  ^iPafo  sin.  aa  -f-  a,  sin.  /3)  sin.  9,  ; 

^2 

and  dividing  by  r^ 

lsr*=-p-i(sin.  «,+-  sin.  /3)sin.  9,- 


i(sin.a,^^  sin.  /3)  sin.  9,  .  .  .  .  (  247). 


*  If  the  direction  of  PI  be  that  of  a  tangent  at  the  point  of  contact  A  of 
the  wheels,  a  case  of  frequent  occurrence,  the  value  of  In  ^vanishing,  that  of  N 
would  appear  to  become  infinite  in  this  expression.  The  difficulty  will  however 
be  removed,  if  we  consider  that  when  aj  becomes,  as  in  this  case,  equal  to  rif 
and  the  point  M  is  supposed  to  coincide  with  A,  Lj  becomes  a  chord  of  the  pitch 


PRESSURES    UPON   WHEELS. 


267 


Substituting  NT/,  for  the  factor,  which  it  represents  in 
equation  (245),  we  have 

PAto  sin.  (d+9)— x  sin.  9 Lpasin.  9,}  — PA  K  sin. 


X  sin.  9  +  — •  sin.  9,}  ='Nr1r.t  sin.  (6  +9) (248). 

Ob 

Solving  this  equation  in  respect  to  P15 

X  sin.  9  H — —sin.  9t 


1  + 


X  sin.  9  +  — —sin.  9. 
a, 


1  — 


TV  sin.  (6  +  9) 


a, 


X  sin.  9  H — —  sin.  9a 

Cbn 


*2  sin. 


"Whence,  performing  actual  division  by  the  denominators  of 
the  fractions  in  the  second  member  of  the  equation,  and 
omitting  terms  of  two  dimensions  in  sin.  915  sin.  <pa,  sin.  9 
(observing  that  !N"  is  already  of  one  dimension  in  those  vari- 
ables), we  have 


r 


circle,  and  is  therefore  represented  by  2ri  sin.  $DBA,  or  2ra  sin.  \  (ai-f-/?) »  s° 
sin.  a\-\ — -  sin.  3 


that 


in.  0      2  sin. 


cos.  !(<Zi-{-/?)_ 


2r1  sin. 


2fi  sin. 


— cos. 


If,  therefore,  we  take  the  angle  ai  =  —  (3,  so  as  to  give  to  PI  the  direction  of 

a  tangent  at  A,  this  expression  will  assume  the  value,  — cos.  0,  or-  ;    so  that 

TI  r 

in  this  case 


sin.  0i  — 


—  ein.  /?  sin. 


268  THE  MODULUS    OF  A   SYSTEM 


cosec.  (4+<puPa+— (249). 


la  this  expression  it  is  assumed  that  the  contact  of  the  teeth 
is  behind  the  line  of  centres. 


219.  THE  MODULUS  OF  A  SYSTEM  OF  TWO  TOOTHED  WHEELS. 

Let  n^  and  n^  represent  the  numbers  of  teeth  in  the 
driving  and  driven  wheels  respectively,  and  let  it  be  ob- 
served that  these  number  are  one  to  another  as  the  radii  of 
the  pitch  circles  of  the  wheels  ;  then,  multiplying  both  sides 

v 
of  equation  (249)  by  at—  ,  we  shall  obtain 


cosec. 


Now  let  A«4>  represent  an  exceedingly  small  increment  of 
the  angle  4>,  through  which  the  driven  wheel  is  supposed  to 
have  revolved,  after  the  point  of  contact  P  has  passed  the 
line  of  centres  ;  and  let  it  be  observed  that  the  first  member 

M          /\-j.  flf* 

of  the  above  equation  is  equal  to  PA—  —  -f  ,  and  that  —  A-^ 

rl  A4»  TI 

represents  the  angle  described  by  the  driving  wheel  (Art. 
204.),  whilst  the  driven   wheel   describes    the    angle  A^; 

whence  it  follows  (Art.  50.)  that  P^J—  A^l  represents  the 

W  ,  / 

work  AlJl  done  by  the  driving  pressure  P1?  whilst  this  angle 
A^  is  described  by  the  driven  wheel, 


cosec. 


Let  now  A-s^  be  conceived  infinitely  small,  so  that  the  first 
member  of  the  above  equation  may  become  the  differential 
co-efficient  of  U,,  in  respect  to  4^.  Let  the  equation,  then, 
be  integrated  between  the  limits  0  and  4/  ;  Pa,  L,,  and  L2. 
and  therefore  ]$"  (equation  247)  being  conceived  to  remain 


OF   TWO   TOOTHED   WHEELS.  269 

constant,  whilst  the  angle  4>  is  described;  we  shall  then 
obtain  the  equation 


cosec.  (6  +<p)   ^+^T  .  S  ____  (250), 

where  S  is  taken  to  represent  the  arc  r^  described  by  the 
pitch  circle  of  the  driven  wheel,  and  therefore  by  that  of  the 
driving  wheel  also,  whilst  the  former  revolves  through  the 
angle  4/. 


220.  THE  MODULUS  OF  A  SYSTEM  OF  TWO  TOOTHED  WHEELS, 

THE  NUMBER  OF  TEETH  ON  THE  DRIVEN  WHEEL  BEING  CON- 
SIDERABLE, AND  THE  WEIGHTS  OF  THE  WHEELS  BEING  TAKEN 
INTO  ACCOUNT. 

It  is  evident  that  the  space  traversed  by  the  point  of  con- 
tact of  two  teeth  on  the  face  of  either  of  them  is,  in  this  case, 
small  as  compared  with  the  radius  of  its  pitch  circle,  and 
that  the  direction  of  the  resultant  pressure  R  (see ^(7.  p.  259.) 
upon  the  teeth  is  very  nearly  perpendicular  to  the  line  01 
centres  BC,  whatever  may  be  the  particular  forms  of  the 
teeth;  provided  only  that  they  be  of  such  forms  as  will 
cause  them  to  act  truly  with  one  another.  In  this  case, 
therefore,  the  angle  BOK  represented  by  d +9  is  very  nearly 

if 
equal  to  -,  and  cosec.  (d  +  <p)=l. 

2i 

Since,  moreover,  RP  is  very  nearly  perpendicular  to  the 
line  of  centres  at  A,  and  that  the  point  of  contact  P  of  the 
teeth  deviates  but  little  from  that  line,  it  is  evident  that  the 
line  AP  represented  by  X  differs  but  little  from  an  arc  of 
the  pitch  circle  of  the  driven  wheel,  and  that  it  differs  the 
less  as  the  supposition  made  at  the  head  of  this  article  more 
nearly  obtains.  Let  us  suppose  -^  to  represent  the  angle 
subtended  by  this  arc  at  the  centre  C  of  the  pitch  circle  of 
the  driven  wheel,  then  will  the  arc  itself  be  represented  by 
TV!',  and  therefore  ^=r^  very  nearly.  Substituting  this 
value  of  X  in  equation  (250),  observing  that  cosec. 

and  that  —  —  —  (equation  227),  and  integrating, 

,   I'iPi     •  ,   I'a 


2YO  INVOLUTE   TEETH. 


.....  (251). 

But  the  driven  or  working  pressure  P2  being  supposed  tc 
remain  constant,  whilst  any  two  given  teeth  are  in  action, 
P3^24^  represents  the  work  u  9  yielded  by  that  pressure  whilst 
those  teeth  are  in  contact  :  also  r^  represents  the  space  S, 
described  by  the  circumference  of  the  pitch  circle  of  either 
wheel  whilst  this  angle  is  described.  Now  let  4/  be  con- 
ceived to  represent  the  angle  subtended  by  the  pitch  of  one 
of  the  teeth  of  the  driven  wheel,  these  teeth  being  supposed 

2* 
to  act  only  behind  the  line  of  centres,  then  ^=  —  ,  ny  repre- 

senting the  number  of  teeth  on  the  driven  wheel,  and  J^ 


:.  Ua=  J  1+  *(-+-)  sin.  9  +  —  sin.  9,  +  —  sin.  <pa  1 
|          \njnj  ay,  a,r9          a( 

U.+  N.  S  .....  (252), 

which  relation  between  the  work  done  at  the  moving  and 
working  points,  whilst  any  two  given  teeth  are  in  contact,  is 
evidently  also  the  relation  between  the  work  similarly  done, 
whilst  any  given  number  of  teeth  are  in  contact.  It  is  there- 
fore the  MODULUS  of  any  system  of  two  toothed  wheels,  the 
numbers  of  whose  teeth  are  considerable. 


221.  THE  MODULUS  OF  A  SYSTEM  OF  TWO  WHEELS  WITH  INVO- 
LUTE  TEETH  OF  ANY  NUMBEES   AND   DIMENSIONS. 

The  locus  of  the  points  of  contact  of  the  teeth  has  been 
shown  (Art.  201.)  to  be  in  this  case 
a  straight  line  DE,  which  passes 
through  the  point  of  contact  A  of 
the  pitch  circles,  and  touches  the 
circles  (EF  and  DG)  from  which  the 
involutes  are  struck.  Let  P  repre- 
sent any  position  of  this  point  of 
contact,  then  is  AP  measured  along 
the  given  line  DE  the  distance  re- 
presented by  X  in  Art.  216.,  and  the 
angle  CAD,  which  is  in  this  case 
constant,  is  that  represented  by  6.  Since,  moreover,  the 
point  of  contact  of  the  teeth  moves  precisely  as  a  point  P 
upon  a  flexible  cord  DE;  unwinding  from  the  circle  EF  and 
winding  upon  DG,  would  (see  note,  p.  238.),  it  is  evident 


INVOLUTE   TEETH. 


271 


that  the  distance  AP,  being  that  which  such  a  point  would 
traverse  whilst  the  pitch  circle  AH  revolved  through  a  cer- 
tain angle  4^  measured  from  the  line  of  centres  is  precisely 
equal  to  the  length  of  string  which  would  wind  upon  DU 
whilst  this  angle  is  described  by  it;  or  to  the  arc  of  that 
circle  which  subtends  the  angle  4>.  If,  therefore,  we  repre- 
sent the  angle  ACD  by  *j,  so  that  CD—  CA  cos.  ACD=ra 
cos.  *),  then  X=?"24'  cos-  i-  Substituting  this  value  for  X  in 

equation  (249),  and  observing  that 


+  <p  =  -  —  *j  -f  <p  =  -  — 


)—  <p),  and  that  —  =  —  ,  we  have 


n 


.....  (253); 


from  which  equation  we  obtain  by  the  same  steps  as  in 
Art.  219,  observing  that  n  is  constant, 


IL—  \ 
( 


—  ) 

n 


cos.  v\  sn. 


sec.   ^- 


(254), 


which  is  the  modulus  of  a  system  of  two  wheels  having  any 
given  numbers  of  involute  teeth. 


222.  THE  ESTVOLTJTE  TOOTH  OF  LEAST  RESISTANCE. 

It  is  evident  that  the  value  of  Uj  in  equation  (254),  or  of 
the  work  which  must  be  done 
upon  the  driving  wheel  to  cause 
a  given  amount  U2  to  be  yielded 
by  the  driven  wheel  is  dependent 
for  its  amount  upon  the  value  of 
the  co-efficient  of  U2  in  the 
second  member  of  that  equation  ; 
and  that  this  co-efficient,  again,  is 
dependent  for  its  value  (other 
things  being  the  same)  upon  the 
value  of  *j  representing  the  angle 
ACD,  or  its  equal  the  angle  DAI, 


272  THE   INVOLUTE  TOOTH   OF   LEAST   RESISTANCE. 

which  the  tangent  DE  to  the  circles  from  which  the  invo- 
lutes are  struck  makes  with  a  perpendicular  AI  to  the  line 
of  centres.  Moreover,  that  the  co-efficient  N  not  involving 
this  factor  ij  (equation  247),  the  variation  of  the  value  of 
[Ju  so  far  as  this  angle  is  concerned,  is  wholly  involved  in 
the  corresponding  variation  of  the  co-efficient  of  U2  and 
becomes  a  minimum  with  it  ;  so  that  the  value  of  i  which 
gives  to  the  function  of  f\  represented  by  this  co-efficient,  its 
minimum  value,  is  the  value  of  it  which  satisfies  the  condi- 
tion of  the  greatest  economy  of  power,  and  determines  that 
inclination  DAI  of  the  tangent  DE  to  the  perpendicular  to 
the  line  of  centres,  and  those  values,  therefore,  of  the  radii 
CD  and  BE  of  the  circles  whence  the  involutes  are  struck, 
which  correspond  to  the  tooth  of  least  resistance. 

To  determine  the  value  of  *\  which  corresponds  to  a  mini- 
mum value  of  this  co-efficient,  let  the  latter  be  represented 
by  u  ;  then,  for  the  required  value  of  *j, 


du 
-r=0, 


-r 
an 


Let  *  (-  +  -}=  A,  ^i£i  sin.  9,  +  —  sin.  93=B  ; 
Vf*j    nj        '  a^  a^ 


cos.  ^  sin.  9  +  B)  sec.  (i—  9); 
sec.  (1—9)4-  A  sin.  9  cos.  ij  sec.  (1—9); 

.*.  -y-  =B  sec.  (q—  9)  tan.  (n—  9)—  A  sin.  9  {  sin.  v  sec.  (*)—  9}— 

cos.  y  tan.  (>i—9)  sec.  (^—9)}  ; 
.    /.  ^r  =B  sec.  \v  —9)  sin.  (*j  —  9)— 
A  sin.  9  sec.  X1?—  9)  {sin.  ^  cos.  (q—  9)—  cos.  »3  sin.  (>)—  -9)}  ; 
.'.  j-=sec.  '(*)—  9){B  sin.  (>)—  <p)—  A  sin.  '9}  .....  (255). 

In  order,  therefore,  that  -r-may  vanish  for  any  value  of 

*i,  one  of  the  factors  which  compose  the  second  member  of 
the  above  equation  must  vanish  for  that  value  of  >j  ;  but 
this  can  never  be  the  case  in  respect  to  the  first  factor,  for 
the  least  value  of  the  square  of  the  secant  of  an  arc  is  the 
square  of  the  radius.  If,  therefore,  the  function  u  admit  of 


THE   INVOLUTE   TOOTH   OF   LEAST   RESISTANCE.  273 

a  minimum  value,  the  second  factor  of  the  above  equation 
vanishes  when  it  attains  that  value ;  and  the  corresponding 
value  of  y  is  determined  by  the  equation, 

B  sin.  (ij  —  9)— A  sin.  a9=0 (256). 

or  by  sin.  (*]—  <p)=  ^-sim  "9  or  by»i=<p+sin.    ( -g-sin.  29J; 
or  substituting  the  values  of  A  and:  B, 

1      1 

(257).. 


yj=9+sn. 


— 'sin.  9r+—  sin.  92 


Now  the  function  u  admits  of  a  minimum,  to>  which  this 
value  of  v  corresponds,  provided  that  when,  substituted  in 

-3-5  this  value  of  *}  gives  to  that  second  differential  co-effi- 

cient of  u  in  respect  to  n  &  positive  value.. 
Differentiating  equation  (255),,  we  have- 


5=2  sec.  "(q  —  9).  tan-,  (i—  9){Bsin.  (>)—  9)- 

A  sin.,  "9}  +B;sec.  a(*i  —  ?)  cos.  (*j—  9) 

But  the  proposed  value-  of  »i  (equation  256)  has  been 
shown  to  be  that  which,,  being  substituted  in  the  factor  {B 
sin.  (>)—  9)—  A  sin.  a9}>  will  cause  it  to  vanish,  and  therefore, 

with  it,  the  whole  of  the  first  term  of  the  value  of  -y-j  :  it 

Ojf\ 

corresponds,  therefore,  to  a  minimum,  if  it  gives  to  the 
second  term  B  sec.  \y  —  9)  cos.  (11  —  9)  a  positive  value  ;  or, 
since  sec.  3(^--9)  is  essentially  positive,  and  B  does  not 
involve  *),  if  it  gives  to  cos.  fy  —  9)  a  positive  value,  or  if 

if  ~~1  /A  \     if  A 

i—  9  <    or  if  sm-  sin-  a<P   <»  or  if      sin.  89<1  ;  or  if 


A  sin.  39  <B  ;  or  if 

sin- 


This  condition  being  satisfied,  the  value  of  tj.  determined 

18 


274          THE   BEST   DIVISION   OF   THE   ANGLE   OF    CONTACT. 

by  equation  (257),  corresponds  to  a  minimum,  and  deter- 
mines the  INVOLUTE  TOOTH  OF  LEAST  RESISTANCE.* 


223.  To  DETERMINE  IN  WHAT  PROPORTION  THE  ANGLE  OF- 
CONTACT  OF  EACH  TOOTH  SHOULD  BE  DIVIDED  BY  THE  LINE 
OF  CENTRES  ;  OR  THROUGH  HOW  MUCH  OF  ITS  PITCH  EACH 
TOOTH  SHOULD  DRIVE  BEFORE  AND  BEHIND  THE  LINE  OF 
CENTRES,  THAT  THE  WORK  EXPENDED  UPON  FRICTION  MAY 
BE  THE  LEAST  POSSIBLE. 

Let  the  proportion  in  which  the  angle  of  contact  of  each 
tooth  is  divided  by  the  line  of  centres  be  represented  by  a?, 

2* 
so  that  x  —  may  represent  the  angular  distance  from  the  line 

ny 

of  centres  of  a  line  drawn  from  the  centre  of  the  driven 
wheel  to  the  point  of  contact  of  the  teeth  when  they  first 

2* 
come  into  action  before  the  line  of  centres,  and  (1— a?)  — 

ni 

the  corresponding  angular  distance  behind  the  line  of  centres 
when  they  pass  out  of  contact ;  and  let  it  be  observed  that, 
on  this  supposition,  if  U2  represent  as  before  the  work 
yielded  by  the  driven  wheel  during  the  contact  of  any  two 
teeth,  a?Ut  will  represent  the  portion  of  that  work  done 
before,  and  (1 — £c)(J2  that  done  behind,  the  line  of  centres. 
Then  proceeding  in  respect  to  equation  (253)  by  the  same 
method  as  was  used  in  deducing  from  that  equation  the 
modulus  (Equation  254),  but  integrating  first  between  the 

limits  0  and  x — ,  in  order  to  determine  the  work  u.  done  by 

< 
the  driving  pressure  before  the  point  of  contact  passes  the 

2* 
line  of  centres,  and  then  between  the  limits  0  and  (1 — x) — 

n* 

to  determine  the  work  u^  done  after  the  point  of  contact  has 
passed  the  line  of  centres  ;  observing  moreover,  that  in  the 
former  case  —9  is  to  be  substituted  in  sec.  (77—9)  for  9  (Art. 
217.),  we  have 

*  It  may  easily  be  shown  by  eliminating  77  between  equations  (254)  and 
(256)  that  the  modulus  corresponding  to  this  condition  of  the  greatest  economy 
of  power,  where  involute  teeth  are  used,  is  represented  by  the  formula 

U1=  |  l-j-*A  sin.  20-f-(B2-A2  sin.  »  I  U2+NS. 


THE   BEST   DIVISION    OF   THE   ANGLE   OF   CONTACT.  275 


-  +  -)cos  *)sin     4-^sm 
n,     nj       '  a,r 


--^sin.  <p2}sec.  (11+9) 


Or  assuming 

*(—  +  —  /cos.  v  sin.  9=0,  and— —  sin.  •<pa  +  -2-i  sin.  <pa=i 
\»,     V  a^  V2 


s1  representing  the  space  described  by  the  pitch  circle  of 
either  wheel  before  the  line  of  centres  is  passed  ;  similarly, 


sec.    11- 


Adding  these  equations  together,  and  representing  by  Uj  the 
whole  work  t^H-if,  done  by  the  driving  pressure  during  the 
contact  of  the  teeth,  and  by  S  the  whole  space  described  by 
the  circumference  of  either  pitch  circle,  we  have 


sec. 


.  .  (259) 

by  which  equation  is  determined  the  modulus  of  two  wheels 
driven  by  involute  teeth,  when  the  contact  takes  place  partly 
before  and  partly  behind  the  line  of  centres. 

Let  the  portion  of  the  work  U^,  which  is  expended  upon 
the  friction  of  the  teeth  be  represented  by  u.     Then 

u=.  I  (ax*  -f-  lx)  sec.  (q  -f  9)  + 


the  value  of  a?,  which  gives  to  this  function  its  mini- 
mum, and  which  therefore  determines  that  division  of  the 
driving  arc  which  corresponds  to  the  greatest  economy  of 
power,  is  evidently  the  value  which  satisfies  the  condition 

^-0  — 

dx~  dx* 

But  differentiating  and  reducing 


276       THE  BEST  DIVISION  OF  THE  ANGLE  OF  CONTACT. 

~r=  \  2##{sec.  (>j+9)  +  sec.  (n—  9)}  -f 

5  {sec.  (*i  +  9)— sec.  Oi— <p)}  —  2#  sec.  (?)— 9)  j-  Ua 
cPu 


Whence  it  appears  that  the  second  condition  is  always  satis- 
fied, and  that  the  first  condition  is  satisfied  by  that  value  of 
a?,  which  is  determined  by  the  equation 

(q-f  9)+sec.  (^—9)}  +5  {sec.  (*]-f  9)—  sec.  (^—9)}  — 


Whence  we  obtain  by  transposition  and  reduction 
#=-  j  1  —  (in  —  I  tan.  if]  tan.  </>  j-  . 

So  that  the  condition  of  the  greatest  economy  of  power  is 
satisfied  in  respect  to  involute  teeth,  when  the  teeth  first 
come  into  contact  before  the  line  of  centres  at  a  point  whose 
angular  distance  from  it  is  less  than  one  half  the  angle  sub- 
tended by  the  pitch  by  that  fractional  part  of  the  last-men- 

tioned angle,  which  is  represented  by  the  formula  -Jjl-j  —  I 

tan.  TJ  tan.  9,  or  substituting  for  J  and  a  their  values  by  the 
formula 


1 

2  " 


i  ,   a^i  «,^ 

~/l — 1\~~  r  tan'  ^  tan*  9  •  • '  ^260)- 


tfi — | — icos.^  sin. 


That  division  of  the  angle  of  contact  of  any  two  teeth  by 
the  line  of  centres,  which  is  consistent  with  the  greatest 
economy  of  power,  is  always,  therefore,  an  unequal  division, 
the  less  portion  being  that  which  lies  before  the  line  of  cen- 
tres ;  and  its  fractional  defect  from  one  half  the  angle  of  con- 
tact, as  also  the  fractional  excess  of  the  greater  portion  above 
one  half  that  angle,  is  in  every  case  represented  by  the  above 
formula,  and  is  therefore  dependent  upon  the  dimensions  of 
the  wheels,  the  forms  and  numbers  of  the  teeth,  and  the  cir- 
cumstances under  which  the  driving  and  working  pressures 
are  applied  to  them.* 

*  The  division  of  the  arc  of  contact  which  corresponds  to  the  greatest  eco- 
nomy of  power  in  epicycloidal  teeth,  may  be  determined  by  precisely  the  same 
steps. 


THE   MODULUS   OF   A   SYSTEM   OF   TWO   WHEELS.  277 


224.  THE  MODULUS  OF  A  SYSTEM  OF  TWO  WHEELS  DRIVEN  BY 
EPIOYCLOIDAL  TEETH. 

The  locus  of  the  point  of  contact  P  of  any  two  such  teeth 
is  evidently  the  generating  circle  APH  of 
the  epicycloidal  face  of  one  of  the  teeth,  and 
the  hypocycloidal  flank  of  the  other  (Art. 
202.)  ;  for  it  has  been  shown  (Art.  199.), 
that  if  the  pitch  circles  of  the  wheel  and  the 
generating  circle  APH  of  the  teeth  be  con- 
ceived to  revolve  about  fixed  centres  B,  C, 
D  by  their  mutual  contact  at  A,  then  will  a 
point  P  in  the  circumference  of  the  last-men- 
tioned circle  move  at  the  same  time  upon 
the  surfaces  of  both  the  teeth  which  are  in 
contact,  and  therefore  always  coincide  with  their  point  of 
contact,  so  that  the  distance  AP  of  the  point  of  contact  P  of 
the  teeth  from  A,  which  distance  is  represented  in  equation 
(250)  by  X,  is  in  this  case  the  chord  of  the  arc  AP,  which 
the  generating  circle,  if  it  revolved  by  its  contact  with 
the  pitch  circles,  would  have  described,  whilst  each  of  the 
pitch  circles  revolved  through  a  certain  angle  measured 
from  the  line  of  centres.  Let  the  angle  which  the  driven 
wheel  (whose  centre  is  C)  describes  between  the  period 
when  the  point  of  contact  P  of  the  teeth  passes  the  line  of 
centres,  and  that  when  it  reaches  the  position  shown  in  the 
figure  be  represented  as  before  by  -^,  the  arc  of  the  pitch 
circle  of  that  wheel  which  passes  over  the  point  A  during  that 
period  will  then  be  represented  by  r^.  ~N ow  the  generating 
circle  APH  having  revolved  in  contact  with  this  pitch  circle, 
an  equal  arc  of  that  circle  will  have  passed  over  the  point  A  ; 
whence  it  follows  that  the  arc  AP=r24' ;  and  that  if  the  radius 
of  the  generating  circle  be  represented  by  r,  then  the  angle 

M 

ADP  subtended  by  the  arc  AP  is  represented  by  —  ^,   or 

7* 

by  2^,  if  20  be  taken  to  represent  the  ratio  —  of  the   radius 

of  the  pi tcli  circle  of  the  driven  wheel  to  the  radius  of  the 
generating  circle.      Now  the  chord  AP=2AD  sin.  £  ADP; 

/Y* 

therefore  \=2r  sin.  e-^——  sin.  e^.     Substituting  this  value 

C? 

of  X  in  equation  (250) ;  observing,  moreover,  that  the  angle 


278  THE   MODULUS   OF   A   SYSTEM    OF   TWO   WHEELS 


PAD  represented  by  6   in  that  equation   is  equal  to  --  -J 

2 

ADP,  or  to  ^—  e\,  and  that  the  whole  angle  4-  through 
which  the  driven  wheel  is  made  to  revolve  by  the  contact  of 
each  of  its  teeth  is  represented  by—  ,  we  have 


sn- 


sn.<p2}sec.  <-< 
tt-fi. 

or,  assuming  Lx  and  L2   to  remain  constant  during  the  con- 

tact of  any  two  teeth  representing  the  constant  1  -f  -^-sin.tpj  -f 

€t1rl 

—  ^sin.  9a  by  A,  and  observing  that  —  =—  , 

27T  2«r 

sn.  e>  sec, 


11,=?^  j  A  /  sec.  (e-\>—  <p)^J/H  —  (lH  —  ^J  sin.  9/  si 
o  '  o 


Now  the  general  integral,  /  sec.  (^—  9)<^4/5  °r 

1   /* 

-  /  sec.  (e^—  $)d(e\—  0)  being  represented*  by  the  function 

£  17 


1     /^cos.(^-0)c?( 
~2e  J      1+sin.  (cV»~ 


rcos.W—tydieii—t)      1.          (  .  .     .     ,   ,      ..  )        1. 

I  v   y  -  1/.         -  -^'rr:  —  lo^       ^  14-  Sin.  (ei/>  —  0)   f  —       log. 

J       1—  sin.  («V-0)         2e     °   e  '  M      2e     6 

,  ,v  I      1,         (  1+sin.  (eV-0)  )  i_ 

1—  sin.  (e^—fy)  \  —-log.   •<  - 

M      e     b  e  (  1—  sin.  (ei/>—  0)  ) 


HAVING   EPICYOLOIDAL   TEETH.  ^  279 

-log.  tan.  \  7+ 4(^—0)  r  >  its  definite  integral  between  the 

6          e  (  *  ) 


limits    0    and  —  has  for  its  expression, 


ff) 

tan.  (-—- 


27T 


s*  74  y»  «2 

Also/  sec.(^— 9)  &in.e^d^==/  sec.(^— 9)  sin.  {(^—9) +9}  d\> 
o  o 

27T 
^^Ti^ 

=/sec.  (e-^— 9)  {sin.  (6^—9)  cos.  9-f-cos.  (e-^— 9)  sin. 
o 

27T 

/T2/2 
jcos.  9  tan.  (e-^— 9)+ sin.  9}^ 
o 

27T 

"1  y^  ^*2  '       ^^ 

= -cos.  9/  tan.  (^—9)  ^  (^4/— 9)  +  —  sin.  9. 


Now  the  general  integral/  tan.  (0^—9)  <f(e^—  9)  has  for 
its  expression—  log.  gcos.  (^—9).*  Taking  its  definite  inte- 
gral between  the  limits  0  and  —  ,  we  have,  therefore, 


2£  /  2^         \ 

/«»  1  cos-\  -  ~V)     2* 

sec.  (e^—  9)  sin.  #4'^=  —  cos.9loff.  _      x  n*        '  H  --  sin.9. 
0  e  /^rka     m  OT. 


COS.  9 


1*1.,  f  JTTi  ^Lxl 

2  sin.  -j  ~~rh  (eip—d)  C  cos.  •)  - — h(ey—<l>  f    j  2 
* ,  i  ^*  '  ^4 

-Ion. 


£  (    2  sin.  |  j_i(^-«  J  cos.  |  J 


280  THE   MODULUS   OF   A   SYSTEM  OF  TWO   WHEELS 

Substituting  these  expressions  in  the  modulus,  representing 
-— -  by  9',  and  observing  that  if  Ua  represent  the  work 

4:      2 

yielded  by  the  driven  wheel  during  the  action  of  each  tooth, 
then  Paaa.?5=Ua,  so  that  Paaa=??2l?,  we  have 

'«*• 

cos.  9 


.  (261). 
cos.  9 

OOS/^—m^  (  20*  )  26* 

JSTow  log.  __V?___/  =  log.e  ]  l+tan.--tan.9  [  cos.--= 
e       cos.  9  a  2 


log.£cos.  —  +log.£  jl  +  tan.— tan.9[=log.£cos.—  + 

tan.— -tan. 9— £tan.a  — —  .  tan.a9  +  &c.      Substituting  this 

expression  in  the  preceding  equation,  and  neglecting  terms 
above  the  first  dimension  in  tan.  9  and  sin.  9, 


cos.  — lJa  +  NS (262). 


26* 

225.  If  the  radius  r  of  the  generating  circle  be  equal  to 
one  half  the  radius  r^  of  the  pitch  circle  of  the  driven  wheel, 
according  to  the  method  generally  adopted  by  mechanics 

(Art.  203.),  then  e=^  =  J—  =1. 
n  *  r         r 

In  this  case,  therefore — that  is,  where  the  flanks  of  the 
driven  wheel  are  straight  (Art.  210.) — the  modulus  becomes 


.  cos.- 


^9 


. 
2*  (        6  *       tan.  9  \       n 

(263). 


HAVING   EPICYCLOIDAL   TEETH.  281 

226.  Substituting  (in  equation  262.)  for  9'  its  value  -  __  - 

4:        2 


lpg-.    tn!  '     =log'< 


**•*  tan.U-i 


If,  therefore,  we  assume  the  teeth  in  the  driven  wheel  to 
be  so  numerous,  or  n^  to  be  so  great  a  number,  that  the  third 

power  and  all  higher  powers  of  tan.  I- -\  may  be  ne- 

\n^      2/ 

glected  as  compared  with  its  first  power,  and  if  we  neglect 

powers  of  tan.  -  above  the  second, 

2 


which  expression  becomes  —  if  we  suppose   the  two  arcs 

which  enter  into  it  to  be  so  small  as  to  equal  their  respec- 
tive tangents. 


! 

Again,  log.g  cos.  —  =  —  %\  —  J  very  nearly.* 

*  For  assume  log.e  cos.  x=alx*-}-a<ix*-\-a*x*-+-  .  .  .  .  ;  then  differentiating, 

—tan.  z=2a1aH-4a2o;34-6a8z6-}-  .....  ; 

2 
but  (Miller,  Diff.  Cal.  p.  95.)—  tan.  x=—  x—  $x*—  —  -x6—  ----  ;  equating, 

therefore,  the  co-efficients  of  these  identical  series,  we  have 


,  ,  3 

x*       x*          2x° 
...log.£COS.*=_T______ 


282  THE   MODULUS    OF   THE   BACK   AND    PINION. 

Substituting  these  values  in  equation  (262),  and  perform- 
ing actual  multiplication  by  the  factor  --2-,  we  have 

-i  +  -)sin.  29 
nl      nj 

or  substituting  for  A  its  value  ;  and  assuming  -J-  sin.  2<p= 
sin.  9,  since  9  is  exceedingly  small, 


n.<p    U,+NS  .....  (264), 

which  is  the  modulus  of  a  wheel  and  pinion  having  epicy- 
cloidal  teeth,  the  number  of  teeth  7i2  in  the  driven  wheel 
being  considerable  (see  equation  252). 

It  is  evident  that  the  value  of  Uj  in  the  modulus  (equa- 
tion 261),  admits  of  a  minimum  in  respect  to  the  value  of  «/ 
there  is,  therefore,  a  given  relation  of  the  radius  of  the 
generating  circle  of  the  driving,  to  that  of  the  driven  wheel, 
which  relation  being  observed  in  striking  the  epicycloidal 
faces  and  the  hypocycloidal  flanks  of  the  teeth  of  two  wheels 
destined  to  work  with  one  another,  those  wheels  will  work 
with  a  greater  economy  of  power  than  they  would  under  any 
other  epicycloidal  forms  of  their  teeth.  This  value  of  e  may 
be  determined  by  assuming  the  differential  co-efficient  of  the 
co-efficient  of  U2  in  equation  (261)  equal  to  zero,  and  solving 
the  resulting  transcendental  equation  by  the  method  of 
approximation. 


227.  THE  MODULUS  OF  THE  BACK  AND  PINION. 

If  the  radius  r9  of  the  pitch  circle  of  the  driven  wheel  be 
supposed  infinite  (Art.  213.),  that  wheel  becomes  a  rack,  and 
the  radius  rl  of  the  driving  wheel  remaining  of  finite  dimen- 
sions, the  two  constitute  a  rack  and  pinion.  To  determine 
the  modulus  of  the  rack  and  pinion  in  the  case  of  teeth  of 
any  form,  the  number  upon  the  pinion  being  great,  or  in 
the  case  of  involute  teeth  and  epicycloidal  teeth  of  any 
number  and  dimensions,  we  have  only  to  give  to  r^  an 
infinite  value  in  the  moduli  already  determined  in  respect 


THE   MODULUS   OF   THE   RACK   AND    PINION.  283 

to  these  several  conditions.     But  it  is  to  be  observed  in 
respect  to  epicycloidal  teeth,  that  n^  becomes  infinite  with 

v 

?*„  whilst  the  ratio—  remains  finite,  and  retains  its  equality 
ni 

to  the  ratio  ^  (equation  227),  so  that^=  -^  =*^.=^  ; 

T 

if  we  represent  the  ratio  —  by  2^.     Making  n^  and  ra  infinite 
in  each  of  the  equations  (252),  (254:),  and  (261),  and  sub- 

P  P 

stituting  —  for  —  in  equation  (262)  ;  we  have 

1.  For  the  modulus  of  the  rack  and  pinion  when  the  teeth 
are  very  small,  whatever  may  be  their  forms,  provided  that 
they  work  truly. 

—'sin.  9x  +  -sm.  9  !  Ua+NS  .....  (265). 

^i^"i  ^i 

2.  For  the  modulus  of  a  rack  and  pinion,  with  involute 
teeth  of  any  dimensions  (see^.  1.  p.  255), 

11!=  ]  1+  I  —  cos.  ??  sin.  <H  —  —sin.  0J  sec.  (rj—  <j>)  [  U3+ 

(.  \W'1  Gbfl*^  I 

JSTS  .  .  (266). 

3.  For  the  modulus  of  the  rack  and  pinion,  with  cycloidal 
and  epicycloidal  teeth  respectively  (equation  261), 

. 


tan.  9' 


20j  cos.  9       ) 

In  each  of  which  cases  the  value  of  N"  is  determined  by 
making  r*  infinite  in  equation  (24:7). 


2)I  (z  +  ^  (!  +i)4  (1  +  l)  =  i.,  becau8e 


e  is  infinite.     The  friction  of  the  rack  upon  its  guides  is  not  taken  into  account 
in  the  above  equations. 


284 


CONICAL  WHEELS. 


CONICAL  OB  BEVIL  WHEELS. 

228.  These  wheels  are  used  to  communicate  a  motion  of 
rotation  to  any  given  axis  from  another,  inclined  to  the  first 
at  any  angle. 

Let  AF  be  an  axis  to  which  a  motion  of  rotation  is  to  be 
communicated  from  another  axis  AE 
inclined  to  the  first  at  any  angle  EAF, 
by  means  of  bevil  wheels. 

Divide  the  angle  EAF  by  the  straight 
line  AD,  so  that  DO  and  DN,  perpen- 
diculars from  any  point  D  in  AD  upon 
AE  and  AF  respectively,  may  be  to 
one  another  as  the  numbers  of  teeth 
which  it  is  required  to  place  upon  the 
two  wheels.* 

Suppose  a  cone  to  be  generated  by  the  revolution  of  the 
line  AD  about  AE,  and  another  by  the  revolution  of  the 
line  AD  about  AF.  Then  if  these  cones  were  made  to 
revolve  in  contact  about  the  fixed  axes  AE  and  AF,  their 
surfaces  would  roll  upon  one  another  along  their  whole  line 
of  contact  DA,  so  that  no  part  of  the  surface  of  one  would 
slide  upon  that  of  the  other,  and  thus  the  whole  surface 
of.  the  one  cone,  which  passes  in  a  given  time  over  the  line 
of  contact  AD,  be  equal  to  the  whole  surface  of  the  other, 
which  passes  over  that  line  in  the  same  time.  For  it  is 
evident  that  if  nv  times  the  circumference  of  the  circle  DP 
be  equal  to  n^  times  that  of  the  circle  DI  and  these  circles 
be  conceived  to  revolve  in  contact  carrying  the  cones  with 
them,  whilst  the  cone  DAP  makes  n^  revolutions,  the  cone 


*  This  division  of  the  angle  EAF  may  be  made  as  follows : — Draw  ST  and 
UW  from  any  points  S  and  U  in  the  straight  lines  AE  and  AF  at  right  angles 
to  those  lines  respectively,  and  having  their 
lengths  in  the  ratio  of  the  numbers  of  teeth 
which  it  is  required  to  place  upon  the  two  wheels ; 
and  through  the  extremities  T  and  W  of  these 
lines  draw  TD  and  WD  parallel  to  AE  and  AF 
respectively,  and  meeting  in  D.  A  straight  line 
drawn  from  A  through  D  will  then  make  the 
required  division  of  the  angle  ;  for  if  DO  and 
DN  be  drawn  perpendicular  to  AE  and  AF,  they 
will  evidently  be  equal  to  UW  and  ST,  and  there- 
fore in  the  required  proportion  of  the  numbers 
of  the  teeth ;  moreover,  any  other  two  lines 
•  drawn  perpendicular  to  AE  and  AF  from  any 

other  point  in  AD  will  manifestly  be  in  the  same  proportion  as  F*0  and  DN. 


CONICAL   WHEELS.  285 

DAI  will  make  n^  revolutions;  so  that  whilst  any  other 
circle  GH  of  the  one  cone  makes  n^  revolutions,  the  corre- 
sponding circle  HK  of  the  other  cone  will  make  n^  revolu- 
tions: but  n^  times  the  circumference  of  the  circle  GH 
is  equal  to  n^  times  that  of  the  circle  HK,  for  the  diameters 
of  these  circles,  and  therefore  their  circumferences,  are  to 
one  another  (by  similar  triangles)  in  the  same  proportion  as 
the  diameters  and  the  circumferences  of  the  circles  DP  and 
DI.  Since,  then,  whilst  the  cones  make  nv  and  n^  revolutions 
respectively,  the  circles  HG  and  HK  are  carried  through  nl 
and  n^  revolutions  respectively,  and  that  n^  times  the  circum- 
ference of  HG  is  equal  to  n^  times  that  of  HK,  therefore 
the  circles  HG  and  HK  roll  in  contact  through  the  whole  of 
that  space,  nowhere  sliding  upon  one  another.  And  the 
same  is  true  of  any  other  corresponding  circles  on  the  cones ; 
whence  it  follows  that  their  whole  surfaces  are  made  to  roll 
upon  one  another  by  their  mutual  contact,  no  two  parts 
being  made  to  slide  upon  one  another  by  the  rolling  of  the 
rest. 

The  rotation  of  the  one  axis  might  therefore  be  communi- 
cated to  the  other  by  the  rolling  of  two  such  cones  in  con- 
tact, the  surface  of  the  one  cone  carrying  with  it  the  surface 
of  the  other,  along  the  line  of  contact  AD,  by  reason  of  the 
mutual  friction  of  their  surfaces,  supposing  that  they  could 
be  so  pressed  upon  one  another  as  to  produce  a  friction  equal 
to  the  pressure  under  which  the  motion  is  communicated,  or 
the  work  transferred.  In  such  a  case,  the  angular  velocities 
of  the  two  axes  would  evidently  be  to  one  another  (equation 
227)  inversely,  as  the  circumferences  of  any  two  correspond- 
ing circles  DP  and  DI  upon  the  cones,  or  inversely  as  their 
radii  ND  and  OD,  that  is  (by  construction)  inversely  as  the 
numbers  and  teeth  which  it  is  supposed  to  cut  upon  the 
wheels. 

When,  however,  any  considerable  pressure  accompanies 
the  motion  to  be  communicated,  the  friction  of  two  such 
cones  becomes  insufficient,  and  it  becomes  necessary  to 
transfer  it  by  the  intervention  of  bevil  teeth.  It  is  the  cha- 
racteristic property  of  these  teeth  that  they  cause  the  motion 
to  be  transferred  by  their  successive  contact,  precisely  as  it 
would  by  the  continued  contact  of  the  surfaces  of  the 
cones. 


286 


CONICAL    WIIICKLS. 


229.  To  describe  the  teeth  of  bevil  wheels.* 

From  D  let  FDE  be  wrawn  at  right  angles  to  AD,  inter- 
secting the  axes  AE  and  AF  of  the  two  cones  in  E  and  F ; 
suppose  conical  surfaces  to  be  generated  by  the  revolution 
of  the  lines  DE  and  DF  about  AE  and  AF  respectively ; 


and  let  these  conical  surfaces  be  truncated  by  planes  LM 
and  XY  respectively  perpendicular  to  their  axes  AE  and 
AF,  leaving  the  distances  DL  and  DY  about  equal  to  the 
depths  which  it  is  proposed  to  assign  to  the  teeth.  Let  now 
the  conical  surface  LDPM  be  conceived  to  be  developed 
upon  a  plane  perpendicular  to  AD,  and  passing  through  the 
point  D,  and  let  the  conical  surface  XIDY  be  in  like 
manner  developed,  and  upon  the  same  plane.  When  thus 
developed,  these  conical  surfaces  will  have  be- 
come the  plane  surfaces  of  two  segmental  annuli 
l&Ppm  and  IXa^f,  whose  centres  are  in  the 
points  E  and  F  of  the  axes  AE  and  AF,  and 
which  touch  one  another  in  the  point  D  of  the 
line  of  contact  AD  of  the  cones. 

Let  now  plane  or  spur  teeth  be  struck  upon 
the  circles  Pj?  and  K,  such  as  would  cause  them 

*  The  method  here  given  appears  first  to  have  been  published  by  Mr.  Tred- 
gold  in  his  edition  of  Buchanan's  Essay  on  Mill-work,  1823,  p.  103. 

f  The  lines  MP  and  pm  in  the  development,  coincided  upon  the  cone,  as  also 
the  lines  IX  and  ix;  the  other  letters  upon  the  development  in  the  above 


CONICAL   WHEELS.  287 

to  drive  one  another  as  they  would  be  driven  by  theii 
mutual  contact ;  that  is,  let  these  circles  Yp  and  Ii  be  taken 
as  the  pitch  circles  of  such  teeth,  and  let  the  teeth  be 
described,  by  any  of  the  methods  before  explained,  so  that 
they  may  drive  one  another  correctly.  Let,  moreover,  their 
pitches  be  such,  that  there  may  be  placed  as  many  such 
teeth  on  the  circumference  P£>  as  there  are  to  be  teeth 
upon  the  bevil  wheel  HP,  and  as  many  on  Ii  as  upon  the 
wheel  III. 

Having  struck  upon  a  flexible  surface  as  many  of  the  first 
teeth  as  are  necessary  to  constitute  a  pattern,  apply  it  to 
the  conical  surface  DLMP,  and  trace  off  the  teetli  from  it 
upon  tli at  surface,  and  proceed  in  the  same  manner  with  the 
surface  DIXY. 

Take  DH  equal  to  the  proposed  lengths  of  the  teeth,  draw 
ef  through  H  perpendicular  to  AD,  and  terminate  the  wheels 
at  their  lesser  extremities  by  concave  surfaces  HGmZ  and 
HKxy  described  in  the  same  way  as  the  convex  surfaces 
which  form  their  greater  extremities.  Proceed,  moreover, 
in  the  construction  of  pattern  teeth  precisely  in  the  same 
way  in  respect  to  those  surfaces  as  the  other ;  and  trace  out 
the  teeth  from  these  patterns  on  the  lesser  extremities  as  on 
the  greater,  taking  care  that  any  two  similar  points  in  the 
teetli  traced  upon  the  greater  and  lesser  extremities  shall  lie 
in  the  same  straight  line  passing  through  A.  The  pattern 
teeth  thus  traced  upon  the  two  extremities  of  the  wheels  are 
the  extreme  boundaries  or  edges  of  the  teeth  to  be  placed 
upon  them,  and  are  a  sufficient  guide  to  the  workman  in 
cutting  them. 


230.  To  prove  that  teeth  thus  constructed  will  work  truly 
with  one  another. 

It  is  evident  that  if  two  exceedingly  thin  wheels  had  been 
taken  in  a  plane  perpendicular  to  AD  (fig.  p.  286.)  passing 

figure  represent  points  which  are  identical  with  those  shown  by  the  game  let- 
ters in  the  preceding  figure.  In  that  figure  the  conical  surfaces  are  shown, 
developed,  not  in  a  plane  perpendicular  to  AD,  but  in  the  plane  which  contains 
that  line  and  the  lines  AE  and  AF,  and  which  is  perpendicular  to  the  last-men- 
tioned plane.  It  is  evidently  unnecessary,  in  the  construction  of  the  pattern 
teeth,  actually  to  develope  the  conical  extremities  of  the  wheels  as  above 
described ;  we  have  only  to  determine  the  lengths  of  the  radii  DE  and  DF  by 
construction,  and  with  them  to  describe  two  arcs,  Pjo,  It,  for  the  pitch  circles 
of  the  teeth,  and  to  set  off  the  pitches  upon  them  of  the  same  lengths  as  the 
pitches  upon  the  circles  DP  and  DI,  which  last  are  determined  by  the  numbeifl 
of  teeth  required  to  be  cut  upon  the  wheels  respectively. 


THE   MODULUS    OF   A   SYSTEM 

through  the  point  D,  and  having  their  centres  in  E  and  F, 
and  if  teeth  had  been  cut  upon  these  wheels  according  to 
the  pattern  above  described,  then  would  these  wheels  have 
worked  truly  with  one  another,  and  the  ratio  of  their  angu- 
lar velocities  have  been  inversely  tkat  of  ED  to  FD,  or  (by 
similar  triangles)  inversely  that  of  ND  to  OD  ;  which  is  the 
ratio  required  to  be  given  to  the  angular  velocities  of  the 
bevil  wheels. 

Now  it  is  evident  that  that  portion  of  each  of  the  conical 
surfaces  DPML  and  DIXY  which  is  at  any  instant  passing 
through  the  line  LY  is  at  that  instant  revolving  in  the  plane 
perpendicular  to  AD  which  passes  through  the  point  D,  the 
one  surface  revolving  in  that  plane  about  the  centre  E,  and 
the  other  about  the  centre  F ;  those  portions  of  the  teeth  of 
the  bevil  wheels  which  lie  in  these  two  conical  surfaces  will 
therefore  drive  one  another  truly,  at  the  instant  when  they 
are  passing  through  the  line  LY,  if  they  be  cut  of  the  forms 
which  they  must  have  had  to  drive  one  another  truly  (and 
with  the  required  ratio  of  their  angular  velocities)  had  they 
acted  entirely  in  the  above-mentioned  plane  perpendicular 
to  AD  and  round  the  centres  E  and  F.  Now  this  is  pre- 
cisely the  form  in  which  they  have  been  cut.  Those  por- 
tions of  the  bevil  teeth  which  lie  in  the  conical  surfaces 
DPML  and  DIXY  will  therefore  drive  one  another  truly  at 
the  instant  when  they  pass  through  the  line  LY ;  and  there- 
fore they  will  drive  one  another  truly  through  an  exceedingly 
small  distance  on  either  side  of  that  line.  Now  it  is  only 
through  an  exceedingly  small  distance  on  either  side  of  that 
line  that  any  two  given  teeth  remain  in  contact  with  one 
another.  Thus,  then,  it  follows  that  those  portions  of  the 
teeth  which  lie  in  the  conical  surfaces  DM  and  DX  work 
truly  with  one  another. 

Now  conceive  the  faces  of  the  teeth  to  be  intersected  by  an 
infinity  of  conical  surfaces  parallel  and  similar  to  DM  and 
DX ;  precisely  in  the  same  way  it  may  be  shown  that  those 
portions  of  the  teeth  which  lie  in  each  of  this  infinite  num- 
ber of  conical  surfaces  work  truly  with  one  another;  whence 
it  follows  that  the  whole  surfaces  of  the  teeth,  constructed  as 
above,  work  truly  together. 


231.  THE  MODULUS  OF  A  SYSTEM  OF  TWO  CONICAL  OB 
BEVIL  WHEELS. 

Let  the  pressure  P,  and  Pa  be  applied  to  the  conical 


OF   TWO   CONICAL    WHEELS.  289 

wheels  represented  in  the  accompanying  figure  at  perpen- 
dicular distances  at  and  #2  from  their  axes  CB  and  CG ;  let 
the  length  AF  of  their  teeth  be  represented  by  & ;  let  the 
distance  of  any  point  in  this  line  from  F  be  represented  by 
a?,  and  conceive  it  to  be  divided  into  an  exceedingly  great 
number  of  equal  parts,  each  represented  by  A#.  Through 
each  of  these  points  of  division  imagine  planes  to  be  drawn 


perpendicular  to  the  axes  CB  and  CG  of  the  wheels,  dividing 
the  whole  of  each  wheel  into  elements  or  laminae  of  equal 
thickness  ;  and  let  the  pressures  P,  and  P2  be  conceived  to  be 

equally  distributed  to  these  laminae.     The  pressure  thus  dis- 

p 
tributed  to  each  will  then  be  represented  bv  —  A#  on  the 

o 
p 
one  wheel,  and  — ?Aa?  on  the  other.     Let^  and^>2  represent 

the  two  pressures  thus  applied  to  the  extreme  laminae  AH 
and  AK  of  the  wheels,  and  let  them  be  in  equilibrium  when 
thus  applied  to  those  sections  separately  and  independently 
of  the  rest ;  then  if  R,  represent  the  pressure  sustained  along 
that  narrow  portion  of  the  surface  of  contact  of  the  teeth  of 
the  wheels  which  is  included  within  these  laminaa,  and  if  R, 
and  R2  represent  the  resolved  parts  of  the  pressure  R  in  the 
directions  of  the  planes  AH  and  AK  of  these  laminae,  the 
pressures^  and  3^  applied  to  the  circle  AH  are  pressures 
in  equilibrium,  as  also  the  pressures  p^  and  R2  applied  to  the 
circle  AK.  If,  therefore,  we  represent  as  before  (Art.  216.) 
by  ml  and  m2,  the  perpendiculars  from  B  and  G  upon  the 

19 


290  THE   MODULUS    OF   A    SYSTEM 

directions  of  K,  and  R2,  and  by  Lt  and  L2,  the  distances  be- 
tween the  feet  of  the  perpendiculars  &„  mt  and  «„,  ma  we 
have  (equation  236,  237),  neglecting  the  weights  of  the 
wheels, 


1  f  /PiLi\  1    • 

l  "&,  +     --  1  sm.  0,  Y   R, 

' 


p!  and  pa  representing  the  radii  of  the  axes  of  the  two  wheels, 
and  9,  and  92  the  corresponding  limiting  angles  of  resistance. 
Let  7j  and  y%  represent  the  inclinations  of  the  direction  of  E 
to  the  planes  of  AH  and  AK  respectively  ;  then 

R!=  R  cos.  715  R2=  R  cos.  ya. 

Now  it  has  been  shown  in  the  preceding  article,  that  the 
action  of  that  part  of  the  surface  of  contact  of  the  teeth  which 
is  included  in  each  of  the  laminse  AH,  AK,  is  identical  with 
the  action  of  teeth  of  the  same  form  and  pitch  upon  two 
cylindrical  wheels  AD  and  AL  of  the  same  small  thickness, 
situated  in  a  plane  EAD  perpendicular  to  AC,  and  having 
their  centres  in  the  intersections,  l>  and  </,  with  that  plane  of 
the  axes  CB  and  CG  produced.  The  reciprocal  pressure  R 
of  the  teeth  of  the  element  has  therefore  its  direction  in  the 
plane  EAD  ;  and  if  its  direction  coincided  with  the  line  of 
centres  DL  of  the  two  circles  EA  and  AD,  then  would  its 
inclinations  to  the  planes  of  AH  and  AK  be  represented  by 
D  AH  and  LAK,  or  by  ACB  and  ACG. 

The  direction  of  R  is  however,  in  overy  case,  inclined  to 
the  line  of  centres  at  a  certain  angle,  which  has  been  shown 
(Art.  216.)  to  be  represented  in  every  position  of  the  teeth, 
after  the  point  of  contact  has  passed  the  line  of  centres  by 
(4  +  9)  j  where  &  represents  the  inclination  to  AL  of  the  line 
X,  which  is  drawn  from  the  point  of  contact  A  of  the  pitch 
circles  to  the  point  of  contact  of  the  teeth,  and  where  9  repre- 
sents the  limiting  angle  of  resistance  between  the  surfaces  of 
the  teeth.  To  determine  the  inclination  /t  of  RA  to  the 
plane  of  the  circle  AH,  its  inclination  RAD  to  the  line  of 
centres  being  thus  represented  by  (0  +  <p),  and  the  inclination 
of  the  plane  AD,  in  which  it  acts,  to  the  plane  AH  being 
DAH,  which  is  equal  to  ACB,  let  this  last  angle  be  repre- 


OF   TWO    CONICAL   WHEELS.  291 

sented  by  i, ;  and  let  Aa  in  the  accom- 
panying figure  represent  the  intersection 
of  the  planes  AD  and  AH  ;  Aard  repre- 
senting a  portion  of  the  former  plane  and 
Aacfi  of  the  latter.  Let  moreover  AT 
represent  the  direction  of  the  pressure  R 
in  the  former  plane  and  let  Ad  and  Ah  be  portions  of  the 
lines  AD  and  AH  of  the  preceding  figure.  Draw  re  per- 
pendicular to  the  plane  Aach,  and  rd  and  ch  parallel  to  A#, 
and  join  dh ;  then  rAc  represents  the  inclination  7,  of  the 
direction  of  R  to  the  plane  AD,  dAr  represents  the  inclina- 
tion (9  +  A)  of  AR-  to  AD,  and  dAh  represents  the  inclination 
ix  of  the  planes  AD  and  AH  to  one  another.  Also,  since  Aa 
is  perpendicular  to  the  plane  AM,  therefore  dr  is  perpen- 
dicular to  that  plane, 

/.  re  =  Ar  sin.  7,  =  Ad  sec.  (6  4-9)  sin.  7^ 
Also  hd  =  Ad  sin.  iiy  but  re  =  hd, 


:.  Ad  sec.  (d-f-9)  sin.  71= 
/.  sin.  7,  —  cos.  (d  +  <p)  sin. 


In  like  manner  it  may  be  shown  that  sin.  72  =  cos.  (d-f  9) 
sin.  ia,  i2  being  taken  to  represent  the  inclination  KAL  of  the 
planes  AE  and  AK,  which  angle  is  also  equal  to  the  angle 
ACG. 

From  the  above  equations  it  follows  that 


!=R  cos.  7,=E  \/l  —  cos.2(d+9)  sin. 
2=R  cos.  72=R  \/l  —  cos.  2(^  +9)  sin. 


From  the  centre  b  of  the  circle  AD  draw  fim  perpendicular 
to  RA,  then  is  BM  (the  perpendicular  let  fall  from  the 
centre  of  the  circle  AH  upon  the  direction  of  R,)  the  projec- 
tion of  lym  upon  the  plane  of  the  circle  AH.  To  determine 
the  inclination  of  bm  to  the  plane  AH,  draw  An  parallel  to 
bm  ;  the  sine  of  the  inclination  of  An  to  the  plane  AH  is 
then  determined  to  be  cos.  DArc/  .  sin.  •„  precisely  as  the  sine 
of  the  inclination  of  Am  to  the  same  plane  was  before  deter- 
mined to  be  cos.  DAm  .  sin.  ie 


Now  T>An=Abm  =  2  —DAR  =  ^  —  (d  +  <p)  ;  therefore  the 
sine  of  the  inclination  of  ATI,  and  therefore  of  bm,  to  the  plane 


292 


THE   MODULUS    OF   A    SYSTEM 


Cv%:: 


AH  is  represented  by  the  formula  sin.  (0+9)  sin.  i,,  and  the 
cosine  of  its  inclination  by  4/1 —sin.  2(0 + 9)  sin.  \  ; 

„•.  m1=BM=5m  i/l— sm.2(d+9)sin.a»1. 

Now  it  has  been  shown  (Art.  216.)  that  the  perpendicular 
~bm  let  fall  from  the  centre  of  a  spur  wheel  upon  the  direc- 
tion of  the  pressure  upon  its  teeth  is,  in  any  position  of  their 
point  of  contact,  represented  (equation  239)  by  the  formula, 


rl  sin. 


sn.  9, 


where  0,  9,  X  represent  the  same  quantities  which  they  have 
been  taken  to  represent  in  this  article  ;  but  rl  represents  the 
radius  5  A  of  the  circle  AD,  instead  of  the  radius  13  A  of  the 
circle  AH;  now  5A=rBA  sec.  DAH=7|1  sec.  ^5  substituting 
this  value  for  r^  in  the  preceding  formula,  we  have 


m=7\  sn.     +<p  sec.  1,+    sn.  9  ; 
/.  ml  =  \T!  sin.  (6  +9)  sec.  ^4-^  sin.  9} 


Similarly  it  may  be  shown  that 

raa  —  \r^  sin.  (6  +  9)  sec.  «2—  X  sin.  9} 


(269). 


Substituting  the  values  of  inl  and  m^  above  determined, 
and  also  the  values  of  Rt  and  R2  (equations  268)  in  equations 


OF  TWO   CONICAL   WHEELS.  293 

(267),  and  eliminating  R  between  those  equations,  a  relation 
will  be  determined  between^?,  and  p^  which  is  applicable  to 
any  distance  of  the  point  of  contact  of  the  teeth  from  the 
line  of  centres. 

Let  it  now  be  assumed  that  the  number  of  the  teeth  of 

the  driven  wheel  is  considerable,  so  that  the  angle  —  tra« 

n* 

versed  by  the  point  of  contact  of  each  tooth  may  be  small, 
and  the  greatest  value  of  the  line  X,  the  chord  of  an  exceed- 
ingly small  arc  of  the  pitch  circle  of  the  driven  wheel.  In 

this  case  d+9  will  very  nearly  equal  -(Art.  220.);   so  that 


cos.  8(^+<p)  will  be  an  exceedingly  small  quantity  and  may 
be  neglected,  and  sin.  (^  +  9)  very  nearly  equal  unity.  Sub- 
stituting these  values  in  equations  (268)  and  (269)  we  have 

K1=E,  R2=R, 

m1=r1+X  sin.  9  cos.  ia,  m2=/'2—  X  sin.  9  cos.  »2. 

Substituting  these  values  in  equations  (26Y)  and  dividing 
those  equations  by  one  another  so  as  to  eliminate  R, 

^4-Xsin.  9  cos.  «,+  I—  1  sin.  9, 
Pi  _  <\  m  _  >  ^1  /  _  . 

1     r^—  Xsin.  9  cos.  <3—  1-^—  M  sin.  9a 

\     $2     / 

x  .  /P,LA 

1  H  —  sm.  9  cos.  »j+         -    sin.  9. 


*i*  •  22       . 

**  1  --  sin.  9  cos.  «„—    -  -    sin.  9, 

rt  \apj 

"Whence  performing  actual  division  by  the  denominator  of 
the  fraction,  and  neglecting  terms  involving  dimensions 
above  the  first  in  sin.  9,  sin.  9,,  sin.  92, 


Now  if  4/  represent  the  angle  described  by  the  driven 
wheel  or  circle  ELA,  whilst  any  two  teeth  are  in  contact, 
since  X  is  very  nearly  a  chord  of  that  circle  subtending  this 
small  angle  4/  (Art.  220.)  ;  /.  X  =  r^.  Let  *  represent  the 


294:  THE   MODULUS    OF   A    SYSTEM. 

angle  described  by  the  conical  wheel  FK,  whilst  the  circle 
ELA  describes  the  angle  ^ ;  then,  since  the  pitch  circle  of 
the  thin  wheel  AK  and  the  circle  ELA  revolve  in  contact  at 
A,  they  describe  equal  arcs  whilst  they  thus  revolve,  respec- 
tively, through  the  unequal  angles  4<  and  *.  Moreover,  the 
radius  Ag  of  the  circle  AL=AG  sec.  GA.g=rt  sec.  *a,  there- 
fore 4^3  sec.  i3=ira; 

.-.,),=*  cos.  if (270). 

Substituting  the  above  valves  of  4'  and  X,  and  observing 

?"o       0*0 
that  —  =  — , 
r,      n. 


L.  <p  + 


Multiplying  both  sides  of  this  equation  by  p^  ain*y  and  ob- 
serving that  p.a.  —  =  p,a,    2A  T ,  and  that  —  A*  is  the  exceed- 
*     n^  l  T^A^'  n^ 

ingly  small  angle  described  by  the  driving  wheel  AN,  whilst 
the  driven  wheel  describes  the  angle  Ai',  so  that  if  A^  repre- 
sent the  work  done  by  the  pressure^  upon  the  lamina  AH, 
whilst  the  angle  A*  is  described  by  the  driven  wheel,  then 

p  d  — A^izrA'w    we  have 
X 

cos.  ^     cos.  <2\ 

1 1  *  cos.  »3  sin.  <p  -f 

n,          nn    I 


or  assuming  Ai'  infinitely  small,  and  integrating  between  the 
limits  0  and  —  (Art.  220.), 


I.     cos.  «2 

cos-  •  8ID- 


sn.  ^  +        •   sn.  9. 


OF   TWO    CONICAL   WHEELS.  295 

Now  the  above  relation  between  the  work  ul  done  by  the 
pressure  pt  upon  the  extreme  element  AH  of  the  driving 
wheel  whilst  any  two  teeth  are  in  contact,  and  the  pressure 
j?2  opposed  to  the  motion  of  the  corresponding  element  of 
the  driven  wheel,  is  evidently  applicable  to  any  other  two 
corresponding  elements  ;  the  values  of  p^  r^  r»  Lj  and  La 
proper  to  those  elements  being  substituted  in  the  formula. 
If,  therefore,  we  represent  by  AU,  that  increment  of  the 
whole  work  Uj  done  upon  the  driving  wheel,  which  is  due 
to  any  one  of  the  elements  into  which  we  have  imagined 
that  wheel  to  be  divided,  and  if  we  substitute  for  j?2  its 

p 
value  -T?Aa?,  assign  to  L15  L2,  r^  r^  their  values  proper  to  that 

element,  and  represent  those  values  by  L,  L',  r,  /, 


or  assuming  A$  infinitely  small,  and  integrating  between  the 

limits  0  and  5,  and  observing  that  P2#2  —   represents  the 

n, 

whole  work  U2  done  upon  the  driven  wheel  under  the  con- 
stant pressure  P2  during  the  contact  of  any  two  teeth, 


iBin.<p  + 


Now  a+x  being  taken  to  represent  the  distance  of  the 
point  of  contact  of  any  two  such  elements  from  C,  and  a  to 
represent  the  distance  OF,  the  radii  r  and  /  of  these  ele- 
ments are  evidently  (by  similar  triangles)  represented  by 

a+x         I         so\  a+x          /         a?\ 

—— r,  or  ^  1  +  -|  /•„  and  —j-ra  or  1 1  +  -  j  rM  r,  and  r,  repre- 
senting the  radii  of  the  extreme  elements  NF  and  OF,  or  of 
the  pitch  circles  of  the  lesser  extremities  of  the  wheels. 
Also  assuming,  as  we  have  done,  the  pressures  R1  and  R9 


296 


THE   MODULUS    OF  A   SYSTEM 


to  be  perpendicular  to  the  lines  BA, 
-ft  GA  joining  the  centre  of  each  ele- 
ment with  their  point  of  contact  A, 
so  that  the  points  M  and  N  (see  fig. 
p.  292.)  coincide  with  the  point  A 
(see  accompanying  figure)*  ;  and  re- 
presenting the  angles  ABD  and  ACE 
made  by  the  perpendiculars  DB  and 
CE  with  the  line  of  centres  by  ^  and 

^respectively;  observing  also  that  AD2=BAa— 2BA.     BD 
cos.  ABD  +  BD5,  so  that  (^-r\  =  1  —  2  (^\  cos.  ABD  + 


(  -  )  ,  we  have,  substituting,  in  the  second  number  of  this 
\-t)A/ 

equation,  for  BA  or  r  its  value  rl  {  1  -f-  -  \ 

\        al 


or  expanding  the  binomials  in  this  expression,  observing 
that  -  is  an  exceedingly  small  quantity,  neglecting  terms 

Ob 

involving  powers   of  that  quantity   above   the  first,   and 
reducing, 


<"». 


Now  Lj  representing  the  value  of  L  when  #=0,  and  6  re 
maining  constant, 


*  The  circles  in  this  figure  represent  two  of  the  corresponding  laminae  into 
which  wheels  have  been  imagined  to  be  divided ;  they  are  not,  therefore,  in  the 
same  planef  Their  planes  intersect  in  AH. 


OF   TWO    CONICAL   WHEELS.  297 

Let  now  the  angle  ADB,  made  in  respect  to  the  first  ele- 
ment of  the  driving  wheel  between  the  perpendicular  BD  or 
#!  and  the  chord  AD  or  L,  be  represented  by  7/1?  and  let  T/, 
represent  the  corresponding  angle  in  the  driven  wheel,  then 

,  T    ,  a 

=  r*        —  '    - 


, 

L,  -  2LA  cos.  77,  +  a?  =  r*,  :.  (—  ' 


L.a. 

-'  cos.  fl.+     r=; 


•         O      1   *  1         I  — -1 1  I      1 1   O  /  _ 

Substituting  these  values  of  I— I  and 2  1— )  I  cos.  d  —  —  1 
in  equation  (273) ; 

\rj  ""  {rj  ~""w  \  y,2  /   \a)  cos<^--  \7r) 


Extracting  the  square  root  of  the  binomial,  and  neglecting 

terms  involving  powers  of  -  above  the  first, 

a 

L       Lx      /  a,  \   lx\  a,  (  L,     x  \ 

—  -.  --  i^j  i_icos>7?=_^  ----  cos.  77.  f  ; 
T       r,       \rj   \a/  rl  (  at      a 


P.  sn.  9. 


0.    ...    pasin.92  fL'  pasin.9a  (  La          5  ) 

Similarly  -^-  /  ?  &  =     ^—  \  -  -  $  -  cos.  r,,  \  . 

Substituting  these  values  in  the  modulus  (equation  272), 

TT       TT    S  -,       /cos-  'i       cos-  '3 

U,  =  U,  ]  1+n—  —  -  +  -  £  cos.  i,  sin.  9  + 

I  \    7ll  7la 


P.sin.^/L,        5 

—  ~   -   -os- 


298  THE   MODULUS    OF   A   SYSTEM 

Now  let  the  angle  BCG,  or  the  inclination  of  the  axes, 
from  one  to  the  other  of  which  motion  is  transferred  by  the 
wheels,  be  represented  by  2< ;  therefore  »1-Ha=2<.  Also  a 
Bin.  ^=7*!  and  a  sin.  <a=ra, 

sin.  i,        T*!       nt 
sin.  »a        7",       7ia  ' 

sin.  \  __  sin.  \  ^       1        cos.  \  _    1       cos.  \ 
1       1 cos.\^cos.%_  /cos. »,     cos.  ia\  /cos.^     cos.  i,\ 

(COS.*.          COS.  ia\  /I    COS.  I.          1\ 
— 1 — J  \—  ^-j-  —  rf]  COS.  ia 

(COS.  <.       COS.  la\                              Wi            ^a 
^H M  cos.  i3  =  ^ ^ . 
n^          n^  1                 \_  cos.  it 

W,  COS.  «a    "      ^a 

^       cos.  i,       cos.  j'+iO,— 01       1— tan. -|(».—».)  tan.  i 
JNow 


cos.  '2  ~~  cos.    i—  i(»,—  »2)  ~  l  +  tan.-J—  «3)tan.  »  J 


y^      sin.  >,      sin.  {t  +  K'i—  0}  _  tan-  '  +  tan.-j-^—  <2) 
al80  n~  sin.  ia~  sin.  j'-K',-',)}  ~~  tan.  i  -  tan.  £(»,-  «a) 


.    cos.< 


cos,          .  yy-M,  A 

JL  ~r~  -  tan.  i 


1   cos.  ',1          1     /     cos.  i,          \ 
,".  ----  a  =:  -  1  ^»  --  n,  ]  = 

w-,  cos.  «a       n        n^  \  2  cos.  «a         V 

—1  (n*—nf)  +  (V—  yQtan^  _ 


OF   TWO   CONICAL   WHEELS.  299 


-     8ec- 1 


.+!)_  (±_1)  tan.',       A+Mcos,,_(!_!\ginV 
n,     nj      \n,     nj  \V  </  \n^    nj 

I  cos.  »!       cos.  a  /I        1  \ 

.     --  -  +  -  -]  cos.  ia  =    --  --    cos.  a«  — 
\    »,  nt    I  *'*t      ***' 

/I        1\   ,  /I         1\        2sin.9i 

I  --  —  1  sin.  '   —  I  —  +  —  I  --  . 
VT^       nj  \n1       nj  n^ 

Substituting  in  the  preceding  relation,  between  Uj  and  Ua, 


which  is  the  modulus  of  the  conical  or  hevil  wheel,  neglecting 
the  influence  of  the  weight  of  the  wheel. 

If  for  cos.  »?,  and  cos.  17,  we  substitute  their  values  (see 
p.  297),  we  shall  obtain  by  reduction 


from  which  equation  it  is  manifest  that  the  most  favourable 
directions  of  the  driving  or  working  pressures  are  those 
determined  by  the  equations 


232.  It  is  evident,  that  if  the  plane  of  the  revolution 
of  such  a  wheel  be  vertical,  the  influence  of  its  weight  must 
be  very  nearly  the  same  as  that  of  a  cylindrical  or  spur 


300      THE  MODULUS  OF  A  SYSTEM  OF  TWO  CONICAL  WHEELS. 

wheel  of  the  same  weight,  having  a  radius  equal  to  the  mean 
radius  of  the  conical  wheel,  and  revolving  also  in  a  vertical 
plane.  If  the  axis  of  the  wheel  be  not  horizontal,  its  weight 
must  be  resolved  into  two  pressures,  one  acting  in  the  plane 
of  the  wheel,  and  the  other  at  right  angles  to  it;  the  latter  is 
effective  only  on  the  extremity  of  the  axis,  where  it  is  borne 
as  by  a  pivot,  so  that  the  work  expended  by  reason  of  it  may 
be  determined  by  Art.  175,  and  will  be  found  to  present 
itself  under  the  form  of  N2  .  S,  where  Na  is  a  constant  and  S 
the  space  described  by  the  pitch  circle  of  the  wheel,  whilst 
the  work  'Ul  is  done.  The  resolved  weight  in  the  plane 
of  the  wheel  must  be  substituted  for  the  weight  of  the  wheel 
in  equation  (247),  which  determines  the  value  of  "N.  Assum- 
ing the  value  of  K",  this  substitution  being  made,  to  be  repre- 
sented by  Nj,  the  whole  of  the  second  term  of  the  modulus 
will  thus  present  itself  under  the  form  (Nt  +  IST2)S. 


,)S  .....  (276). 

233.  Comparing  the  modulus  of  a  system  of  two  conical 
wheels  with  that  of  a  system  of  two  cylindrical  wheels 
(equation  252),  it  will  be  seen  that  the  fractional  excess 
of  the  work  U2  lost  by  the  friction  of  the  latter  over  that 
lost  by  the  friction  of  the  former  is  represented  by  the 
formula 

2*  sin.  2<  sin,  <p      .,  5  /p.  p.  \ 

+  -    -j-     -+  t-  (-cos.  ^  sin.  9a  +  -  cos.??2sin.<p2j  .  .  . 

The  first  term  of  this  expression  is  due  to  the  friction  of 
the  teeth  of  the  wheels  alone,  as  distinguished  from  the  fric- 
tion of  their  axes  ;  the  latter  is  due  exclusively  to  the  friction 
of  the  axes.  Both  terms  are  essentially  positive,  since  ^ 

and  %  are  in  every  case  less  than  -. 

2 

Thus,  then,  it  appears  that  the  loss  of  power  due  to  the 
friction  of  bevil  wheels  is  (other  things  being  the  same) 
essentially  less  than  that  due  to  the  friction  of  spur  wheels, 
so  that  there  is  an  economy  of  power  in  the  substitution  of 


THE   MODULUS    OF   A   TRAIN    OF   WHEELS.  301 

a  bevil  for  a  spur  wheel  wherever  such  substitution  is  prac- 
ticable. This  result  is  entirely  consistent  with  the  experience 
of  engineers,  to  whom  it  is  well  known  that  bevil  wheels  run 
lighter  than  spur  wheels. 


234.  THE  MODULUS  OF  A  TRAIN  OF  WHEELS. 

In  a  train  of  wheels  such  as  that  shown  in  the  accompany- 
ing figure,  let  the  radii  of  their 
pitch  circles  be  represented  in 
order  by  r»  r»  r3  .  .  .  rt,  begin- 
ning from  the  driving  wheel ; 
and  let  a^  represent  the  perpen- 
dicular distance  of  the  driving 
pressure  from  the  centre  of  that 
wheel,  and  a^  that  of  the  driven 
pressure  or  resistance  from  the  centre  of  the  last  wheel  of  the 
train  ;  U,  the  work  done  upon  the  first  wheel,  uy  the  work 
yielded  by  the  second  wheel  to  the  third,  us  that  yielded  by 
the  fourth  to  the  fifth,  &c.,  and  U2  the  work  yielded  by  the 
last  or  nth  wheel  upon  the  resistance,  then  is  the  relation  be- 
tween Uj  and  ut  determined  by  the  modulus  (equation  252), 
it  being  observed  that  the  point  of  application  of  the  resist- 
ance on  the  second  wheel  is  its  point  of  contact  5  with  the 
third  wheel,  so  that  in  this  case  a^=^rs. 

These  substitutions  being  made,  and  L2  being  taken  to 
represent  the  distance  between  the  point  Z>  and  the  projection 
of  the  point  a  upon  the  third  wheel,  we  have 

11,=  j  1+W-+-W  9+  ^  sin.  9,+ 
(          \n,    nj  a,rt 

^ Bin. ?  j  «,+*«.  8,. 

To  determine,  in  like  manner,  the  relation  between  u^  and 
u^  or  the  modulus  of  the  third  and  fourth  wheels,  let  it  be 
observed  that  the  work  u^  which  drives  the  third  wheel  has 
been  considered  to  be  done  upon  it  at  its  point  of  contact  b 
with  the  fourth  ;  so  that  in  this  case  the  distance  between  the 
point  of  contact  of  the  driving  and  driven  wheels  and  the 
foot  of  the  perpendicular  let  fall  upon  the  driving  pressure 
from  the  centre  of  the  driving  wheel  vanishes,  and  the  term 

*  See  note  p.  266. 


302  THE   MODULUS   OF   A    TRAIN    OF    WHEELS. 

which  involves  the  value  of  L  representing  that  line  disap- 
pears from  the  modulus,  whilst  the  perpendicular  upon  the 
driving  pressure  from  the  centre  of  the  driving  wheel  be- 
comes rs.  Let  it  also  be  observed,  that  the  work  of  the 
fourth  wheel  is  done  at  the  point  of  contact  c  of  the  fifth  and 
sixth  wheels,  so  that  the  perpendicular  upon  the  direction  of 
that  work  from  the  axis  of  the  driven  wheel  is  /•,.  We  shall 
thus  obtain  for  the  modulus  of  the  third  and  fourth  wheels, 


In  which  expression  L3  represents  the  distance  between  the 
point  G  and  the  projection  of  the  point  &  upon  the  fifth 
wheel. 

In  like  manner  it  may  be  shown,  that  the  modulus  of  the 
fifth  and  sixth  wheels,  or  the  relation  between  u3  and  u#  is 

»  •  s>  ; 


and  that  of  the  seventh  and  eighth  wheels,  or  the  relation 
between  u4  and  u# 


<=  \ 


and  that,  if  the  whole  number  of  wheels  be  represented  by 
2p.  or  the  number  of  pairs  of  wheels  in  the  train  by  p,  then 
is  the  modulus  of  the  last  pair, 


In  which  expressions  the  symbols  !N",,  !N",,  N8  .  .  .  "Np  ,  are 
taken  to  represent,  in  respect  to  the  successive  pairs  of  wheels 
of  the  train,  the  values  of  that  function  (equation  247), 
which  determines  the  friction  due  to  the  weights  of  those 
wheels  ;  and  each  of  the  symbols  L,,  Ls,  L4  .  .  .  1^  ,  the  dis- 
tance between  the  point  of  contact  of  a  corresponding  pair 
of  wheels  and  the  projection  upon  its  plane  of  the  point  of 
contact  of  the  next  preceding  pair  in  the  train  ;  whilst  the 
symbols  n^  n^  ns  .  .  .  n^p  ,  represent  the  numbers  of  teeth  in 
the  wheels  ;  r^  r»  r^  .  .  .  r^P  ,  the  radii  of  their  pitch  circles  ; 
and  S15  Sa,  S8  .  .  .  Sp  ,  the  spaces  described  by  their  points  ot 


THE   MJDULTIS   OF   A   TRAIN    OF   WHEELS.  303 

contact  a,  5,  c,  &c.  whilst  the  work  TJj  is  done  upon  the  first 
wheel  of  the  train. 

Let  us  suppose  the  co-efficients  of  i*a,  u^  u±  .  .  .  Ua,  in  these 
moduli  to  be  represented  by  (l+f\),  (l+M-2)>  (1+f*,)  .  .  .  . 
(1-f  fly)  ;  they  will  then  become 


Eliminating  wa,  w8,  %4  ...%,,  between  these  equations,  we 
shall  obtain  an  equation  of  the  form 


.  S  .  .  .  (2TT), 
where 


p  .....  (278). 

Now  let  it  be  observed,  that  the  space  described  by  the  first 
wheel,  at  distance  unity  from  its  centre,  whilst  the  space  Sx 

Q 

is  described  by  its  circumference,  is  represented  by  —  ,  and 

r, 

o 

that  this  same  space  is  represented  by  —  if  S  represent  the 

<*i 

space  described  in  the  same  time  by  the  foot  of  the  per- 
pendicular «15  or  the  space  through  which  the  moving 
pressure  may  be  conceived  to  work  during  that  time  ;  so 

that  —  =    .      Also  let  it   be  observed  that  the   space  de- 
fi     ai 

scribed  by  the  third  wheel,  at  distance  unity  from  its  centre, 
is  the  same  with  that  described  at  the  same  distance  from 

S      S 

its   centre   by   the    second    wheel,  so   that    —  =—  ;  in  like 

rs     r, 

manner  that  the  spaces  described  at  distances  unity  from 
their  centres  by  the  fourth  and  fifth  wheels  are  the  same,  so 

Q  C  Q          Q 

that  —  ——  ;    and  similarly,   that  —  =—  ,   &c.=&c.  ;     and 
^     r,  r,     rt' 

finally, 


304  THE   MODULUS    OF   A   TRAIN   OF   WHEELS. 

Multiplying  the  two  first  of  these  equations  together,  then 
the  three  first,  the  four  first,  &c.,  and  transposing,  \ve  have 


,    ., 

a,  0,  .  r,        \aj  \n,l 


,    n,  .  m, 

g     r..n.n.Ag     /M  (!'8, 

0,  .  ra  .  r4  .  T-.        \V  \ 
&c.=&c. 


Substituting  these  values  of  S,,  Sa,  &c.  in  equation  (278), 
and  dividing  by  S,  we  have 


or  if  we  observe  that  the  quantities  i^,,  ^3,  f*3,  are  composed 
of  terms  all  of  which  are  of  one  dimension  in  sin.  9,  sin.  <p1? 
sin.  92,  &c.  and  that  the  quantities  !N"15  N3,  N",,  &c.  (equation 
247)  are  all  likewise  of  one  dimension  in  those  exceedingly 
small  quantities  ;  and  if  we  neglect  terms  above  the  first 
dimension  in  those  quantities,  then 


'-*  •••}•••»">• 


If  in  like  manner  we  neglect  in  equation  (277)  terms  of 
more  than  one  dimension  in  M-a,  f*a,  ^8,  &c.  we  have 


Now  1^=  *  (  —  j  __  )  sin.  9  +  —  ^  sin.  9,  +  —2^  si 

\n,       nj  a,r,  r.r, 

af  =  if  I  --  [_  —  )  sin.  9  +  —^  sin.  90 

\/i8       nj  r4rt 


.  S. 
sin. 


THE   MODULUS   OF    A   TRAIN   OF   WHEELS.  305 

/*  =  if  (—  +  —  )  sin.  9  +  —  ^  sin.  <p4, 
\n^  nj  r0r, 

&c.=&c. 


=  «r  /-!_  +  —  \  sin.  9  +  5^  si 
W,P-I  '     n,p  ]  r,pa, 


Substituting  these  values  of  f^,  f*a,  &c.  in  the  preceding 
equation, 


jL+.L+-L  ----  L\  sin.9  +  ^  sin.?! 
nl     nt    n£         n,p  J  a.r., 


which  is  a  general  expression  for  the  modulus  of  a  train  of 
any  number  of  wheels. 


235.  The  work  IT,  which  must  be  done-  upon  the  first 
wheel  of  a  train  to  yield  a  given  amount  U3.at  the  last  wheel, 
exceeds  the  work  IJ2,  or,  in  other  words,  the  work  done  upon 
the  driving  point  exceeds-  that  yielded  at  the  working  point, 
by  a  quantity  which  is  represented  by  the  expression 

—  +  --  h  .  ..-  +  -  iwiM'.IL-fi-^  sm-  9  H  —  —sin.  9, 
^  r  T 


.  .  .  (281). 

In  which  expression  the  first  term  represents  the  expenditure 
of  work  due  to  the  friction  of  the  teeth,  and  varies  directly  as 
the  work  Ua,  which  is  done  by  the  machine.  The  second 
term  represents  the  expenditure  of  work  due  to  the  friction 
of  the  axes  of  the  wheels,  and  varies  in  like  manner  directly 
as  the  work  done.  Whilst  the  third  term  represents  the 
expenditure  of  work  due  to  the  weights  of  the  wheels  of  the 
train,  and  is  wholly  independent  of  the  work  done,  but  only 
upon  the  space  S,  through  which  that  work  is  done  at  the 
point  where  the  driving  pressure  is  applied  to  the  train. 


20 


306  FRICTION   OF   THE   AXES   OF   A   TKAIN. 


236.  The  expenditure  of  work  due  to  the  friction  of  the  teeth. 

The  work  expended  upon  the  friction  of  the  teeth  is  repre- 
sented by  the  formula 

r  j( h  -  -  +  —  +...+ }  sin.  9 (282), 

whose  value  is  evidently  less  as  the  factor  sin.  9  is  less,  or  as 
the  coefficient  of  friction  between  the  common  surfaces  of  the 
teeth  is  less  ;  and  as  the  numbers  of  the  teeth  in  the  different 
wheels  which  compose  the  train  are  greater.  The  number 
of  teeth  in  any  one  wheel  of  the  train  may,  in  fact,  be  taken 
so  small,  as  to  give  this  formula  a  considerable  value  as  com- 
pared with  U2,  or  to  cause  the  expenditure  of  work  upon  the 
friction  of  the  teeth  to  amount  to  a  considerable  fraction  of 
the  work  yielded  by  the  train :  and  the  numbers  of  teeth 
of  two  or  more  wheels  of  such  a  train  might  even  be  taken 
so  small  as  to  cause  the  work  expended  upon  their  friction  to 
equal  or  to  surpass  by  any  number  of  times  the  work  yielded 
by  the  train  at  its  working  point.  This  will  become  the 
more  apparent  if  we  consider  that  the  surfaces  of  contact  of 
the  teeth  of  wheels  are  for  the  most  part  free  from  unguent 
after  they  have  remained  any  considerable  time  in  action,  so 
that  the  limiting  angle  of  resistance  assumes  in  most  cases 
a  much  greater  value  at  the  surfaces  of  the  teeth  of  the 
wheels  than  at  their  axes.  From  this  consideration  the 
importance  of  assigning  the  greatest  possible  number  of 
teeth  to  the  wheels  of  a  train  individually  and  collectively 
is  apparent. 


23V.  The  expenditure  of  work  due  to  the  friction  of  the  axes. 
This  expenditure  is  represented  by  the  formula 


forming  the  second  term  of  formula  280.  Now,  evidently,  the 
value  of  this  formula  is  less  as  the  quantities  sin.  9,,.  sin.  9,, 
&c.  are  less,  or  ae  the  limiting  angles  of  resistance  between 
the  surfaces  -of  the  axes  and  their  bearings  are  less,  or  the 


FRICTION   OF   THE   AXES    OF   A   TRAIN.  307 

lubrication  of  the  axes  more  perfect ;  and  it  is  less  as  the 

fractions^-',    ^,    ^,  Ac.  are  less. 

flyy    ya7y    P4jy 

Now,  La  being  the  distance  between  the  point  of  contact  1> 

of  the  third  and  fourth  wheels 
and  the  projection  of -the  point  of 
contact  a  of  the  first  and  second 
upon  the  plane  of  those  wheels, 
it  follows  that,  generally,  L2  is 
least  when  the  projection  of  a 
falls  on  the  same  side  of  the  axis 
as  the  point  5  ;*  and  that  it 
is  least  of  all  when  this  line  falls  on  that  side  and  in  the  line 
joining  the  axis  with  the  point  5;  whilst  it  is  greatest  of  all 
when  it  falls  in  this  line  produced  to  the  opposite  side  of  the 
axis.  In  the  former  case  its  value  is  represented  by  /•,—/•„ 
and  in  the  latter  by  ^t+ya ;  so  that,  generally,  the  maximum 
and  minimum  values  of  La  are  represented  by  the  expression 

y3±y2,  and  the  maximum  and  minimum  values  of    — ^-2-    by 

*•*• 

JL |  p2.    And  similarly  it  appears  that  the  maximum  and 

r       rj 

minimum  values  of  — ^  are  represented  by( 1 )  P3 ;     and 

TV  \T  T   I 

»  '  4  '  5  V/4  '  6  ' 

so  of  the  rest.  So  that  the  maximum  and  minimum  values  of 
the  work  lost  by  the  friction  of  the  axes  are  represented 
by  the  expression 


from  which  expression  it  is  manifest,  that  in  every  case  the 
expenditure  of  work  due  to  the  friction  of  the  axes  is  less  as 
the  radii  of  the  axes  are  less  when  compared  with  the  radii 
of  the  wheels ;  being  wholly  independent  of  actual  dimensions 
of  these  radii,  but  only  upon  the  ratio  or  proportion  of  the 
radius  of  each  axis  to  that  of  its  corresponding  wheel :  more- 

*  Thte  important  condition  is  but  a  particular  case  of  the  general  principle 
established  in  Art.  168. ;  from  which  principle  it  follows,  that  the  driving 
pressure  on  each  wheel  should  be  applied  on  the  same  side  of  the  axis  as  the 
driven  pressure. 


308  THE    WEIGHTS    OF    THE    WHEELS. 

over,  that  this  expenditure  of  work  is  the  least  when  the 
wheels  of  the  train  are  so  arranged,  that  the  projection  of  the 
point  of  contact  of  any  pair  upon  the  plane  of  the  next 
following  pair  shall  lie  in  the  line  of  centres  of  this  last  pair, 
between  their  point  of  contact  and  the  axis  of  the  driving 
wheel  of  the  pair  ;  whilst  the  expenditure  is  greatest  when 
this  projection  falls  in  that  line  but  on  the  other  side  of  the 
axis.  The  difference  of  the  expenditures  of  work  on  the 
friction  of  the  axes  under  these  two  different  arrangements 
of  the  train  is  represented  by  the  formula 

A-  sin.  <P!  +  —  sin.  9,  +  —  sin.  <p3  -f  ASm.<p4  +  .  .  i  U  • 
r,  rs  r&  r,  ) 


which,  in  a  train  of  a  great  number  of  wheels,  may  amount 
to  a  considerable  fraction  of  U2  ;  that  fraction  of  Ua  repre- 
senting the  amount  of  power  which  may  be  sacrificed  by  a 
false  arrangement  of  the  points  of  contact  of  the  wheels. 


238.  The  expenditure  of  work  due  to  the  weights  of  the  several 
wheels  of  the  train. 

The  third  and  last  term  !N" .  S  of  the  expression  (280)  repre- 
sents the  expenditure  of  work  due  to  the  weights  of  the 
several  wheels  of  the  train;  of  this  term  the  factor  1ST  is 
represented  by  an  expression  (equation  -279),  each  of  the 
terms  of  which  involves  as  a  factor  one  of  the  quantities  ~N19 
N3,  Ns,  <fec.,  whose  general  type  or  form  is  that  given  in 
equation  (247),  it  being  observed  that  the  direction  of  the 
driving  pressure  on  any  pair  of  the  wheels  being  supposed 
that  of  a  tangent  to  their  point  of  contact ;  the  case  is  that 
discussed  in  the  note  to  page  266.  The  other  factor  of  each 
term  of  the  expression  (equation  279)  for  !N",  is  a  fraction 
having  the  product  na  n&  .  .  .  of  the  numbers  of  teeth  in  all 
the  preceding  drivers  of  the  train,  except  the  first,  for  its 
numerator,  and  the  product  ny .  n4 .  n6  .  .  ,  of  the  numbers  of 
teeth  in  the  preceding  followers  of  tlie  train  for  its  denomi- 
nator ;  so  that  if  the  train  be  one  by  which  the  motion  is  to 
be  accelerated,  the  numbers  of  teeth  in  the  followers  being 
small  as  compared  with  those  in  the  drivers,  or  if  the  multi- 
plying power  of  the  train  be  great,  and  if  the  qiiantities 
Kj,  N9,  N3,  &c.,  be  all  positive ;  then  is  the  expenditure  of 
work  by  reason  of  the  weights  of  the  wheels  considerable,  as 


MODULUS  OF  A  TRAIN  IN  WHICH  THE  DRIVEKS  ARE  EQUAL.       309 

compared  with  the  whole  expenditure.  Since,  moreover,  the 
coefficients  of  N13  Na,  !N",,  &c.,  in  the  expression  for  N  (equa- 
tion 279)  increase  rapidly  in  value,  this  expenditure  of  work 
is  the  greatest  in  respect  to  those  wheels  of  the  train  which 
are  farthest  removed  from  its  first  driving  wheel :  for  which 
reason,  especially,  it  is  advisable  to  diminish  the  weights 
of  the  wheels  as  they  recede  from  the  driving  point  of  the 
train,  which  may  readily  be  done,  since  the  strain  upon  each 
successive  wheel  is  less,  as  the  work  is  transferred  to  it  under 
a  more  rapid  motion. 


239.  The  modulus  of  a  tram  in  which  all  the  drivers  cure 
equal  to  one  another  and  all  the  followers,  and  in  which 
the  points  of  contact  of  the  drivers  and  followers  are  all 
similarly  situated. 

The  numbers  of  teeth  in  the  drivers  of  the  train  being  in 
this  case  supposed  equal,  and  also  the  radii  of  these  wheels, 
n1=  ns=n6=n1=&c.j  r1=r3=rf>=?\=&c.  The  numbers  of 
teeth  in  the  followers  being  also  equal,  and  also  the  radii  of 
the  followers  n^n^n^&c.,  r^—r^r^&c. 

If,  moreover,  to  simplify  the  investigation,  the  driving 

work  U1  be  supposed  to  be  done  upon  the  first  wheel  of  the 

9  train  at  a  point  situated  in  re- 

spect to  the  point  of  contact  a  of 
that  wheel  with  its  pinion  pre- 
cisely as  that  point  of  contact  is 
in  respect  to  the  point  of  contact 
I  of  the  next  pair  of  wheels  of 
the  train  ;  and  if  a  similar  sup- 
position be  made  in  respect  to 
the  point  at  which  the  driven  work  Ua  is  done  upon  the  last- 
pinion  of  the  train,  then,  evidently,  L,=L2=:L3—  .  .  .  =LP, 
and  (see  equation  247)  1^=^",=  .  .  .  =NP. 

The  modulus  (equation  280)  will  become,  these  substitu- 
tions being  made  in  it,  the  axes  being,  moreover,  supposed 
all  to  be  of  the  same  dimensions  and  material,  and  equally 
lubricated,  and  it  being  observed  that  the  drivers  and  the 
followers  are  each  p  in  number, 


----  (284), 
which  is  the  modulus  required. 


% 
310  THE   TRAIN    OF   LEAST   RESISTANCE. 

Moreover,  the  value  of  !N"  (equation  27Y)  will  become  by 
the  like  substitutions, 


THE  TRAIN  OF  LEAST  RESISTANCE. 

240.  A.  train  of  equal  driving  wheels  and  equal  followers 
being  required  to  yield  at  the  last  wheel  of  the  train  a 
given  amount  of  work  U2,  under  a  velocity  m  times  greater 
or  less  than  that  under  which  the  work  U,  which  drives  the 
train  is  done  by  the  moving  power  upon  the  first  wheel;  it 
is  required  to  determine  what  should  be  the  number  p  of 
pairs  of  wheels  in  the  train,  so  that  the  work  TJ,  expended 
through  a  given  space  S,  in  driving  it,  may  be  a  minimum. 

Since  the  number  of  revolutions  made  by  the  last  wheel 
of  the  train  is  required  to  be  a  given  multiple  or  part  of  the 
number  of  revolutions  made  by  the  first  wheel,  which  mul- 
tiple or  part  is  represented  by  m,  therefore  (equation  231), 


_ 
:.  —  =  — ,  and  —  =  — ; 

'          *  ' 


Substituting  these  values  in  the  modulus  (equation  284); 
substituting,  moreover,  for  N  its  value  from  equation  (285), 
we  have 


THE   TKAIN   OF   LEAST   RESISTANCE.  311 


(286X 


It  is  evident  that  the  question  is  solved  by  that  value  of  p 
which  renders  this  function  a  minimum,  or  which  satisfies 

the  conditions  -j-2  =  0  and  -^  >  0.      The    first    condition 
dp  dp    ' 

gives  by  the  differentiation  of  equation  (286), 
Iog.E 


(m— 


This  equation  may  be  solved  in  respect  to  p,  for  any  given 
values  of  the  other  quantities  which  enter  into  it,  ly  approxi- 
mation. If,  being  differentiated  a  second  time,  the  above 
expression  represents  a  positive  quantity  when  the  value  of 
p  (before  determined)  is  substituted  in  it,  then  does  that 
value  satisfy  both  the  conditions  of  a  minimum,  and  sup- 
plies, therefore,  its  solution  to  the  problem. 

If  we  suppose  91=0  and'N^O,  or,  in  other  words,  if  we 
neglect  the  influence  of  the  friction  of  the  axes  and  of  the 
weights  of  the  wheels  of  the  train  upon  the  conditions  of  the 
question,  we  shall  obtain 


, 

—  gin.  9  H  —  sin.  9=0  ; 
p       n^  7i, 

whence  by  reduction, 

p  =  ^-'mi*  ......  (288). 


*  This  formula  was  given  by  the  late  Mr.  Davis  Gilbert,  in  his  paper  on  the 
"  Progressive  improvements  made  in  the  efficiency  of  steam  engines  in  Corn- 
wall," published  in  the  Transactions  of  the  Royal  Society  for  1830.  Towards 
the  conclusion  of  that  paper,  Mr.  Gilbert  has  treated  of  the  methods  best 
adapted  for  imparting  great  angular  velocities,  and,  in  connection  with  that 
subject,  of  the  friction  of  toothed  wheels ;  having  reference  to  the  friction  of 
the  surfaces  of  their  teeth  alone,  and  neglecting  all  consideration  of  the  influ- 


THE  INCLINED  PLANE. 


THE  INCLINED  PLANE. 

241.  Let  AB  represent  the  surface  of  an  inclined  plane  on 
which  is  supported  a  body  whose  centre  of  gravity  is  C,  and 
its  weight  W ,  by  means  of  a  pressure  acting  in  any  direction, 
and  which  may  be  supposed  to  be  supplied  by  the  tension  of 
a  cord  passing  over  a  pulley  and  carrying  at  its  extremity 
a  weight. 

Let  OR  represent  the  direction  of  the  resultant  of  P  and 
W.  If  the  direction  of  this  line  be  inclined  to  the  perpen- 
dicular ST  to  the  surface  of  the  plane,  at  an  angle  OST 
equal  to  the  limiting  angle  of  resistance,  on  that  side  of  ST 
which  is  farthest  from  the  summit  B  of  the  plane  (as  in 
Jig.  1),  the  body  will  be  upon  the  point  of  slipping  upwards; 
and  if  it  be  inclined  to  the  perpendicular  at  an  angle  OST, 

ence  due  to  the  weights  of  the  wheels  and  to  the  friction  of  their  axes.  The 
author  has  in  vain  endeavoured  to  follow  out  the  condensed  reasoning  by  which 
Mr.  Gilbert  has  arrived  at  this  remarkable  result ;  it  supplies  another  example 
of  that  rare  sagacity  which  he  was  accustomed  to  bring  to  the  discussion  of 
questions  of  practical  science.  Mr.  Gilbert  has  given  the  following  examples 
of  the  solution  of  the  formula  by  the  method  of  approximation:— If  ra=120, 
or  if  the  velocity  is  to  be  increased  by  the  train  120  times,  then  the  value  of  p 
given  by  the  above  formula,  or  the  number  of  pairs  of  wheels  which  should 
c  impose  the  train,  so  that  it  may  work  with  a  minimum  resistance,  reference 
being  had  only  to  the  friction  of  the  surfaces  of  the  teeth,  is  3-745 ;  and  the  value 

<f  the  factor  p(mp  -f-1)  (equation  286),  which  being  multiplied  by —  sin.  (f>  Ua 

HI 

i  ^presents  the  work  expended  on  the  friction  of  the  surfaces  of  the  teeth,  is  in 
this  case  17'9 ;  whereas  its  value  would,  according  to  Mr.  Gilbert,  be  121  if  the 
velocity  were  got  up  by  a  single  pair  of  wheels.  So  that  the  work  lost  by  the 
friction  of  the  teeth  in  the  one  case  would  only  be  one  seventh  part  of  that  in 
the  other.  In  like  manner  Mr.  Gilbert  found,  that  if  m=100,  then  p  should 
equal  3*6 ;  in  which  case  the  loss  by  friction  of  the  teeth  would  amount  to  the 
sixth  part  only  of  the  loss  that  would  result  from  that  cause  if  _p=l,  or  if  the 
required  velocity  were  got  up  by  one  pair  of  wheels. 

If  m=40,  then  jt)=2'88,  with  a  gain  of  one  third  over  a  single  pair. 

If  7tt=3-69,  thenjt>=l. 

If  ra=12'85,  then  p=2. 

If  ra=46'3,  then  jt>=3. 

If  m=166'4,  thenjo=4. 

It  is  evident  that  when  p,  in  any  of  the  above  examples,  appears  under  the 
form  of  a  fraction,  the  nearest  whole  number  to  it,  must  be  taken  in  practice. 
The  influence  of  the  weights  of  the  wheels  of  the  train,  and  that  of  the  friction 
of  the  axes,  so  greatly  however  modify  these  results,  that  although  they  are 
fully  sufficient  to  show  the  existence  in  every  case  of  a  certain  number  of 
wheels,  which  being  assigned  to  a  train  destined  to  produce  a  given  accelera- 
tion of  motion  shall  cause  that  train  to  produce  the  required  effect  with  the 
least  expenditure  of  power,  yet  they  do  not  in  any  case  determine  correctly 
what  that  number  of  wheels  should  be. 


THE   INCLINED   PLANE. 


313 


(i.) 


equal  to  the  limiting  angle  of  resistance,  but  on  the  side  of 
ST  nearest  to  the  summit  B  (as  in  fig.  2.),  then  the  body  will 
be  upon  the  point  of  slipping  downwards  (Art.  138.)  ;  the 
former  condition  corresponds  to  the  superior  and  the  latter 
to  the  inferior  state  bordering  upon  motion  (Art.  140.). 

Now  the  resistance  of  the  plane  is  equal  and  opposite  to 
the  resultant  of  P  and  "W  ;  let  it  be  represented  by  K. 

There  are  then  three  pressures  P,  W,  and  K  in  equili- 
brium. 


sn. 


Let  /BAC=i,  ZOST=:lime.  Z  of  resistance  =9,  let  0 
represent  the  inclination  PQB  of  the  direction  of  P  to  the 
surface  of  the  plane,  and  draw  OY  perpendicular  to  AB  ; 
then, 


mfig.  1, 
and  POK 

in  fig.  2.,  WOR=WOY-SOY=BAC-OST=i--9, 
and 


=PQB+£-OST=^+*-9  ; 

2  a 


and 


the  upper  or  lower  sign  being  taken  according  as  the  body 
is  upon  the  point  of  sliding  up  the  plane,  as  in  fig.  1,  or 
down  the  plane,  as  in  fig.  2.  Or  if  we  suppose  the  angle  9 
to  be  taken  positively  or  negatively  according  as  the  body  is 
on  the  point  of  slipping  upwards  or  downwards  ;  then  gene- 


rally WOK=«+<p 


314 


THE   INCLINED   PLANE. 


P         sin.  (1  +  9)      __sin.  (»+<p)  t 

"W"~~~        lie          ~\~ cos.  (0—9  ' 
sin.  (-+0-9) 


sn- 


*  cos.(d—  9) 


(289). 


If  the  direction  of  P  be  parallel  to  the  plane,  /  PQB  01 
1=0  ;  and  the  above  relation  becomes 


If  i=0  the  plane  becomes  horizontal  (fig.  3).,  and  the  re- 
lation between  P  and  W  assumes  the  form 


P=W 


sin.  9 


—  9) 


(291). 


If  0=0,  P=W  .  tan.  9,  as  it  ought  (see  Art.  138.). 

If  the  angle  PQB  or  &  (fig.  1.)  be  increased  so  as  to  be- 
come «—  d,  rQ  will  assume  the  direction  shown  in  fig.  4, 
and  the  relation  (equation  289),  between  P  and  "W  will  be- 


come 


sn. 


(292). 


cos.  (d  +  <p)  ' 

The  negative  sign  showing  that  the  direction  of  P  must, 
in  order  that  the  body  may  slip  up  the  plane,  be  opposite  to 
that  assumed  in  fig.  1. ;  or  that  it  must  be  a  pushing  pres- 
sure in  the  direction  PO  instead  of  a  pulling  pressure  in  the 
direction  OP. 

If,  however,  the  body  be  upon  the  point  of  slipping  down 
the  plane,  so  that  9  must  be  taken  negatively  ;  and  if,  more- 
over, 9  be  greater  than  i,  then  sin.  0  +  9),  will  become  sin. 
(1—9)^=— sin.  (9—1),  so  that  P  will  in  this  case  assume  the 
positive  value 


. 
cos.  (d—  9) 


(293), 


THE    MOVEABLE   INCLINED    PLANE. 


315 


which  determines  the  force  just  necessary  under  these  cir- 
cumstances to  pull  the  body  down  the  plane. 

If  i=<p,  P— 0,  the  body  will  therefore,  in  this  case,  be  upon 
the  point  of  slipping  down  the  plane  without  the  application 
of  any  pressure  whatever  to  cause  it  to  do  so,  other  than  its 
own  weight.  The  plane  is  under  these  circumstances,  said 
to  be  inclined  at  the  angle  of  repose,  which  angle  is  there- 
fore equal  to  the  limiting  angle  of  resistance. 


242.  The  direction  of  least  traction. 

Of  the  infinite  number  of  different  directions  in  which  the 
pressure  P  may  be  applied,  each  requiring  a  different  amount 
to  be  given  to  that  pressure,  so  as  to  cause  the  body  to  slide 
up  the  plane,  that  direction  will  require  the  least  value  to  be 
assigned  to  P  for  this  purpose,  or  will  be  the  direction  of 
least  traction,  which  gives  to  the  denominator  of  the  fraction 
in  equation  (289)  its  greatest  value,  or 
which  makes  0—9=0  or  d=<p.  The  di- 
rection of  P  is  therefore  that  of  least 
traction  when  the  angle  PQB  is  equal  to 
the  limiting  angle,  a  relation  which  ob- 
tains in  respect  to  each  of  the  cases  dis- 
cussed in  the  preceding  article. 


243.  THE  MOVEABLE  INCLINED  PLANE. 

Let  ABC  represent  an  inclined 
plane,  to  the  back  AC  of  which 
is  applied  a  given  pressure  P,, 
and  which  is  moveable  between 
the  two  resisting  surfaces  GH  and 
KL,  of  which  either  remains  fixed, 
and  the  other  is  upon  the  point 
of  yielding  to  the  pressure  of  the 
plane. 

If  we  suppose  the  resultants  of  the  resistances  upon  the 
different  points  of  the  two  surfaces  AB  and  BG  of  the  plane 
to  be  represented  by  T^  and  R3  respectively,  it  is  evident 
that  the  directions  of  these  resistances  and  of  the  pressure  Pt 


316  THE  MOVEABLE  INCLINED  PLANE. 

will  meet,  when  produced,  in  the  same  point  O*  ;  and  that, 
since  the  plane  is  upon  the  point  of  slipping  upon  each  of 
the  surfaces,  the  direction  of  each  of  these  resistances  is 
inclined  to  the  perpendicular  to  the  surface  of  the  plane,  at 
the  point  where  it  intersects  it,  at  an  angle  equal  to  the  cor- 
responding limiting  angle  of  resistance. 

So  that  if  ET  and  FS  represent  perpendiculars  to  the 
,  surfaces  AB  and  BC  of  the  plane  at  the  points  E  and  F  and 
91?  9a,  the  limiting  angles  of  resistance  between  these  surfaces 
of  the  plane  and  the  resisting  surfaces  GH  and  KL  re- 
spectively, then  K.ET^,  K2FS=9?. 

Now  the  pressures  P15  3^,  R2  being  in  equilibrium  (Art. 
W), 

P,     sin.  EOF  P^sin.  EOF 

K^sin.DOF'  £    *  K3-~sin.DOE' 

But  the  four  angles  of  the  quadrilateral  figure   BEOF 
being  equal  to  four  right  angles  (Euc.  1'32),  EOF—  2*— 


EBF-OEB-OFB;    but   EBF=«,   OEB+9,     OFB= 


Similarly,  DOE=2rr-ADO-AEO-DAE;  but  ADO  = 
J  AEO=~?l,  BAC=^-«:   .-.DOE=J+.+9l. 

Since,  moreover,  DO  is  parallel  to  BC,  both  being  per- 
pendicular to   AC,   /.DOF=tf—  OFC;  but  OFC=^—  <p,  : 


P.  _sin.  {«•—((+  9,  +  9,)}     sin.  Q  +  9.  +  9.)  . 


sn. 


cos.  92 
t_  sin,  {if—  Q  +  9i  +  <p9){  _sin. 


Since  either  is  equal  and  opposite  to  the  resultant  of  the  other  two. 


A  SYSTEM  OF  TWO  MOVEABLE  INCLINED  PLANTS.  317 


cos. 


In  the  case  in  which  the  surface  GH  yields  to  the  pressure 
of  the  plane,  KL  remaining  fixed,  we  obtain  (equation  121.) 
for  the  modulus  (see  Art.  148.)  observing  that  P1(°)=K1  sin.  i 
(equation  294), 

u  sin.  0+9,+yJ  _  (29g)_ 

sm.  i  .  cos.  <pa 

In  the  case  in  which  the  surface  KL  yields,  CH  remaining 
fixed,  observing  that  P1(°)=Tla  tan.  i  (equation  295),  we  have 


sn. 

(297)* 


Equations  (296)  and  (297)  may  be  placed  respectively  un- 
der the  forms 


and  U  =U  CQS>  (<p*+(p*  J  tan.i  +  tan.(91+9.)  1 

sin.  j        (  cot.  91—  tan.  •)  tan.  i  )  * 

The  value  of  II,  corresponding  to  a  given  value  of  U3  is  in 

the  former  equation  a  minimum  when  »=-,  and  in  the  latter 

2t 

when 


tan.  i=.  \  A/-. C°!'<P;    ,     x-l  1  tan.  (9, +9.) ....  (298). 

f  l  sin.  <p1sm.(<p1+<p,)         ) 

From  the  former  of  these  equations  it  follows,  that  the  work 
lost  by  friction  (when  the  driving  surface  of  the  plane  is  its 
hypothenuse)  is  less  as  the  inclination  of  the  plane  is  greater, 
or  as  its  mechanical  advantage  is  less. 


244.  A  system  of  two  moveable  inclined  planes. 
Let  A  and  B  represent  two  inclined  planes,  of  which  A 


318  A  SYSTEM  OF  TWO  MOVEABLE  INCLINED  PLANES. 

P  rests  upon  a  horizontal  surface,  and 

receives  a  horizontal  motion  from 
the  action  of  the  pressure  P, ;  com- 
municating to  B  a  motion  which  is 
restricted  to  a  vertical  direction  by 

the  resistance  of  the  obstacle  D, 

^  \          ~^        which  vertical  motion  of  the  plane 

^T3 i\.    """          *s   °PP°sed  by  the  pressure  P2  ap- 
*  plied  to  its  superior  surface.     It  is 

required  to  determine  a  relation  between  the  pressures  Px 
and  P2,  in  their  state  bordering  upon  motion  ;  and  the  mo- 
dulus of  the  machine. 

Let  R1  represent  the  pressure  of  the  plane  A  upon  the 
plane  B,  or  the  resistance  of  the  latter  plane  upon  the  former, 
and  R3  the  resistance  of  the  obstacle  D  upon  the  back  of  the 
plane  B  ;  then  is  the  relation  between  Rj  and  P,  determined 
by  equation  (294).  And  since  R1?  R3,  P2  are  pressures  in 
equilibrium,  the  relation  between  R1  and  P2  is  expressed 


(Art.   14.)  by  the  relation^-1  =sm'  \      Now    R3Q    is 


inclined  to  a  perpendicular  to  the  back  of  the  plane  B,  at  an 
angle  equal  to  the  limiting  angle  of  resistance  between  the 
surface  of  that  plane  and  the  obstacle  D  on  which  it  is  upon 
the  point  of  sliding.  Let  this  angle  be  represented  by  9,, 
then  is  the  inclination  of  R3  to  the  back  of  the  plane  or  P2Q 

represented  by^—  93  ;  so  that  P2QR3—  -—  9,. 

And  if  R8Q  be  produced  so  as  to  meet  the  surface  of  the 
plane  A  in  Y,  and  YS    be  drawn  horizontally, 


where  i  represents  the  inclination  of  the  superior  surface  of 
the  plane  A  or  the  inferior  surface  of  the  plane  B  to  the 
horizon.  Substituting  these  values  of  PaQRs  and  R^R,,  we 
obtain 


cos.<p8 


Multiplying  this  equation  by  equation  (294),  and  solving  in 
respect  to  Pj, 


A   SYSTEM   OF   THREE   INCLINED    PLANES. 


319 


COS.  (t  +  P.  +  9.)  COS.  <f, 


.-.  (Art.  152.)  U,=U, 


sn. 


,)  cos.?s 


cos. 


3)  tan.  t  cos.<?2 


_  (goo) 


A  system  of  three  inclined  planes,  two  of  which  are  movea- 
lie,  and  the  third  fixed. 

245.  The  inclined  plane  A,  in  the  accompanying  figure,  is 
fixed  in  position,  the  plane  B  is 
moveable  upon  A,  having  its  upper 
surface  inclined  to  the  horizon  at  a 
less  angle  than  the  lower  ;  and  C  is 
an  inclined  plane  resting  upon  B, 
which  is  prevented  from  moving 
horizontally  by  the  obstacle  D,  but 
may  be  made  to  slide  along  this 
obstacle  vertically.  It  is  required 
to  determine  a  relation  between 
Pj  and  Pa,  applied,  as  shown  in  the  figure,  when  the  system 
is  in  the  state  bordering  upon  motion. 

Let  Rj,  R2,  R3  represent  the  resistances  of  the  surfaces  on 
which  motion  takes  place,  <pa  <P2  93  their  limiting  angles  of 
resistance  respectively,  and  il9  £2  the  inclinations  of  the  two 
surfaces  of  contact  of  B  to  the  horizon.  Since  P0  K1?  R2  are 
pressures  in  equilibrium,  as  also  P2,  E2,  K8 

.   P}__sin1R2OR1     R2     sin.P,QR3 
'  R2~  sin.  P.OIV    P2-  sin.  R2QR8- 


Multiplying  these  equations  together, 

P1_sin.EaOB1.8in.PilQB, 
P2  —  sin.  P/JK,  .  sin.  R2QR3 


Draw  OS  and  OT  parallel  to  the  faces  of  the  plane  B  ;  then 


-TOS;  but  R.OS^  --<?,,  since  OS  is 

parallel  to  the  inferior  face  of  the  plane  B,  also  QOT=  —  <pa, 

2 
since  OT  is  parallel  to  the  superior  face  of  the  plane  B  ;  and 


320  A   SYSTEM   OF   THREE   INCLINED   PLANES. 

TOS  =  the  inclination  of  the  faces  of  the  plane  B  to  one 
another  =  i1—  »a. 


/.K^OK^  (2  -<P> 
Also  P9QK3=  I  -K3QM=  |  -  9.. 

LetP.O  be  produced  to  Y;  therefore  P1OKl=*'-K1OY= 

^-(I^OS-SOV)^-  j  g  ~9,)  -i,  |  =  I  +  i,  +  9,-  Lastly 
K2QK3  =  OQM+MQK3'.  Now,  MQK3=93  ;  also,  OQM  = 
*-(QOT+TOY)=*-  j  (—9.)  +',}  =J-',+9« 

.%  E9QE3=r    —i,  +9,+98=    —(',—9,—  9.). 


a      sin.  (s+'i  +  Pi)  •  8^n-  ]  o  ~(t*—V*—V*)  f 
,  __p  sin.  j(91 + 9a)  +  Qi — Q}  cos.  98 


Whence  we  obtain  for  the  modulus  (Art.  152.),  observing; 
that^<»)=8in-('~''). 

COS. «,  COS.  I, 

TT  ^-rr  sin.  (9,+9,+^-Qcos.^cos.  i  cos.9, 


THE  WEDGE  DRIVEN  BY   PRESSURE. 


321 


THE  WEDGE  DRIVEN  BY  PRESSURE. 

246.  Let  ACB  represent  an  isosceles  wedge,  whose  angle 
ACB  is  represented  by  2*,  and  which  is 
driven  between  the  two  resisting  surfaces 
DE  and  DF,  by  the  pressure  P,.  Let  R, 
and  R2  represent  the  resistances  of  these 
surfaces  upon  the  acting  surfaces  CA  and 
CB  of  the  wedge  when  it  is  upon  the 
point  of  moving  forwards.  Then  are  the. 
directions  of  R,  and  R2  inclined  respec- 
tively to  the  perpendicular  Gs  and;  R£' 
to  the  faces  C A  and  CB  of  the  wedge^  at 
angles  each  equal  to  the  limiting  angle  of 
resistance  9.  The  pressures  Rr  and  Ra  are 
therefore  equally  inclined  to  the  axis 
of  the  wedge,  and  to  the  direction;  of  I\,  whence  it  follows 
that  R^E,,,  and  therefore  (Art.  13;)  that  P1=2R1  cos.  £GOR. 
Now,  since  CGOE  is  a  quadrilateral  figure,  its  four  angles 
are  equal  to  four  right  angles ;  therefore  GOR=2*— GCR— 

OGC— ORC.    But  GCR=2i;  OGC^ORC  =  £  +9 : 

a 


(303). 


Whence  it  follows  (equation  121)  that  the  modulus  of  the 
wedge  is 

.  (304). 


sn.  i 


This  equation  may  be  placed  under  the  form 
11!=  U3  jcot.  9  -f-  cot.  i\  sin.  9. 

The  work  lost  by  reason  of  the  friction  of  the  wedge  is 
greater,  therefore,  as  the  angle  of  the  wedge  is  less;  and 
infinite  for  a  finite  value  of  9,  and  an  infinitely  small  value 
of*. 


The  angle  of  the  wedge. 

247.  Let  the  pressure  P0  instead  of  being  that  just  suffi- 
21 


322 


THE   WEDGE    DRIVEN   BY   PRESSURE. 


cient  to  drive  tlie  wedge,  be  now  supposed 
to  be  that  which  is  only  just  sufficient  tc 
keep  it  in  its  place  when  driven.  The  two 
surfaces  of  the  wedge  being,  under  these 
circumstances,  upon  the  point  of  sliding 
u  backwards  upon  those  between  which  the 
wedge  is  driven,  at  their  points  of  contact 
G  and  R,  it  is  evident  that  the  directions 
of  the  resistances  if*  and  i^R  upon  those 
points,  must  be  inclined  to  the  normals 
6-G  and  tH  at  angles,  each  equal  to  the 
limiting  angle  of  resistance,  but  measured 
on  the  sides  of  those  normals  opposite  to 

those  on  which  the  resistances  RjG  and  R2R  are  applied.* 
In  order  to  adapt  equation  (303)  to  this  case,  we  have 

only  then  to  give  to  9  a  negative  value  in  that  equation.     It 

will  then  become 


Pl=2E1sin.(«—  9) 


(305). 


So  long  as  i  is  greater  than  9,  or  the  angle  C  of  the  wedge 
greater  than  twice  the  limiting  angle  of  resistance,  Pl  is 
positive  ;  whence  it  follows  that  a  certain  pressure  acting  in 
the  direction  in  which  the  wedge  is  driven,  and  represented 
in  amount  by  the  above  formula,  is,  in  this  case,  necessary 
to  keep  the  wedge  from  receding  from  any  position  into 
which  it  has  been  driven.  So  that  if,  in  this  case,  the  pres- 
sure Pj  be  wholly  removed,  or  if  its  value  become  less  than 
that  represented  by  the  above  formula,  then  the  wedge  will 
recede  from  any  position  into  which  it  has  been  driven,  or 
it  will  be  started.  If  i  be  less  than  9,  or  the  angle  C  of  the 
wedge  less  than  twice  the  limiting  angle  of  resistance,  P, 
will  become  negative  ;  so  that,  in  this  case,  a  pressure,  oppo- 
site in  direction  to  that  by  which  the  wedge  has  been  driven, 
will  have  become  necessary  to  cause  it  to  recede  from  the 
position  into  wThich  it  has  been  driven  ;  whence  it  follows, 
that  if  the  pressure  P,  be  now  wholly  removed,  the  wedge 
will  remain,  fixed  in  that  position  ;  and,  moreover,  that  it 
will  still  remain  fixed,  although  a  certain  pressure  be  applied 
to  cause  it  to  recede,  provided  that  pressure  do  not  exceed 
the  negative  value  of  P1?  determined  by  the  formula. 

*  This  will  at  once  be  apparent,  if  we  consider  that  the  direction  of  the 
resultant  pressure  upon  the  wedge  at  G  must,  in  the  one  case,  be  such,  that  if 
it  acted  alone,  it  would  cause  the  surface  of  the  wedge  to  slip  downwards  on 
the  surface  of  the  mass  at  that  point,  and  in  the  other  case  upwards  ;  and  that 
the  resistance  of  the  mass  is  in  each  case  opposite  to  this  resultant  pressure. 


THE   WEDGE   DRIVEN   BY   IMPACT.  323 

It  is  this  property  of  remaining  fixed  in  any  position  into 
which  it  is  driven  when  the  force  which  drives  it  is  removed, 
that  characterises  the  wedge,  and  renders  it  superior  to 
every  other  implement  driven  by  impact. 

It  is  evidently,  therefore,  a  principle  in  the  formation  of  a 
wedge  to  be  thus  used,  that  its  angle  should  be  less  than 
twice  the  limiting  angle  of  resistance  between  the  material 
which  forms  its  surface,  and  that  of  the  mass  into  which  it 
is  to  be  driven. 


THE  WEDGE  DRIVEN  BY  IMPACT. 

248.  The  wedge  is  usually  driven  by  the  impinging  of  a 
heavy  body  with  a  greater  or  less  velocity  upon  its  back,  in 
the  direction  of  its  axis.  Let  W  represent  the  weight  of 
such  a  body,  and  V  its  velocity,  every  element  of  it  being 
conceived  to  move  with  the  same  velocity.  The  work 
accumulated  in  this  body,  when  it  strikes  the  wedge,  will 

then  be  represented  (Art.  66.)  by  -  —  V2.    Now  the  whole  of 

$ 
this  work  is  done  by  it  upon  the  wedge,  and  by  the  wedge 

upon  the  resistances  of  the  surfaces  opposed  to  its  motion  ; 
if  the  bodies  are  supposed  to  come  to  rest  after  the  impact, 
and  if  the  influence  of  the  elasticity  and  mutual  compression 
of  the  surfaces  of  the  striking  body  and  of  the  wedge  are 
neglected,  and  if  no  permanent  compression  of  their  surfaces 

1  W  V3 

follows  the  impact.*      .*.  Uj  =  -    -  . 

2  9 

*  The  influence  of  these  elements  on  the  result  may  be  deduced  from  the 
principles  about  to  be  laid  down  in  the  chapter  upon  impact.  It  results  from 
these,  that  if  the  surfaces  of  the  impinging  body  and  the  back  of  the  wedge, 
by  which  the  impact  is  given  and  received,  be  exceedingly  hard,  as  compared 
with  the  surfaces  between  which  the  wedge  is  driven,  then  the  mutual  pressure 
of  the  impinging  surfaces  will  be  exceedingly  great  as  compared  with  the 
resistance  opposed  to  the  motion  of  the  wedge.  Now,  this  latter  being 
neglected,  as  compared  with  the  former,  the  work  received  or  gained  by  the 
wedge  from  the  impact  of  the  hammer  will  be  shown  in  the  chapter  upon 

impact  to  be  represented  by  *•   ~r~e'  -  1  -  -  —  ,   where   Wj    represents    the 

" 


weight  of  the  hammer,  W2  the  weight  of  the  wedge,  and  e  that  measure  of 
'the  elasticity  whose  value  is  unity  when  the  elasticity  is  perfect.  Equating 
this  expression  with  the  value  of  Ui  (equation  304),  and  neglecting  the  effect 
of  the  elasticity  and  compression  of  the  surfaces  G  and  R,  between  which  the 
wedge  is  driven,  we  shall  obtain  the  approximation 

_  (l-f-g)3W1aWaVa       sin,  i 
sin. 


324  THE  'WEDGE   DRIVEN   BY   IMPACT. 

Substituting  this  value  of  Ul  in  equation  303,  and  solving  in 
respect  to  U2,  we  have 


sn..      ..... 

2     g     sin.  (*  4-  9) 

by  which  equation  the  work  U2  yielded  upon  the  resistances 
opposed  to  the  motion  of  the  wedge  by  the  impact  of  a  given 
weight  "W  with  a  given  velocity  Y  is  determined  ;  or  the 
weight  "W  necessary  to  yield  a  given  amount  of  work  when 
moving  with  a  given  velocity  ;  or,  lastly,  the  velocity  Y  with 
which  a  body  of  given  weight  must  impinge  to  yield  a  given 
amount  of  work. 

If  the  wedge,  instead  of  being  isosceles,  be  of  the  form  of 
a  right  angled  triangle,  as  shown 
in  the  accompanying  figure,  the 
relation  between  the  work  U,  done 
upon  its  back,  and  that  yielded 
upon  the  resistances  opposed  to 
its  motion  at  either  of  its  faces,  is 
represented  by  equations  (296) 
and  (297).  Supposing  therefore 
this  wedge,  like  the  former,  to  be 


driven  by  impact,  substituting  as  before  for  Uj  its  value 

1"W 

-  —  Y2,  and  solving  in  respect  to  Ua,  we  have,  in  the  c 

which  the  face  AB  of  the  wedge  is  its  driving  surface 


sn 


2     g      si 
when  the  base  BC  of  the  wedge  is  its  driving  surface, 

q.  __1  WY2    tan,  i  cos. 
8     2     g     '   8in. 


From  this  expression  it  follows,  that  the  useful  work  is  the  greatest,  other 
things  being  the  same,  when  the  weight  of  the  wedge  is  equal  to  the  weight 
of  the  hammer,  and  when  the  striking  surfaces  are  hard  metals,  so  that  the 
value  of  e  may  approach  the  nearest  possible  to  unity. 


THE  MEAN  PBESStTEE  OF  IMPACT. 


325 


I 


249.  If  the  power  of  th6  wedge 
be  applied  by  the  intervention  of 
an  inclined  plane  moveable  in  a 
direction  at  right  angles  to  the  di- 
rection of  the  impact*,  as  shown  in 
the  accompanying  figure,  then  sub- 
stituting for  U,  in  equation  (300) 


respect  to  U2,  we  have 

cos.  (i  -h 9t  +9,)  tan.  <  cos.  9a 
cos  98 


half  the  vis  viva  of  the  impinging 
body,  and  solving,  as  befo 


ore,  in 


sn. 


(309). 


If  instead  of  the  base  of  the 
plane  being  parallel  to  the  direc- 
tion of  impact,  it  be  inclined  to 
it,  as  shown  in  the  accompanying 
figure,  then,  substituting  as  above 
in  equation  302,  we  have 


a—  9,)  cos. 


sn.  (t1— 


3     2     g       sin.  fo 


i—  «2  cos.  i,  cos.  *2eos.<p8 


^^ 

'  ' 


THE  MEAN  PKESSUKE  OF  IMPACT. 

250.  It  is  evident  from  equations  306,  307,  308,  that,  since, 
whatever  may  be  the  weight  of  the  impinging  body  or  the 
velocity  of  the  impact,  a  certain  finite  amount  of  work  U2  is 
yielded  upon  the  resistances  opposed  to  the  motion  of  the 
wedge  ;  there  is  in  every  such  case  a  certain  mean  resistance 
H  overcome  through  a  certain  space  S,  in  the  direction  in 
which  that  resistance  acts,  which  resistance  and  space  are 
such,  that 

KS=U2,  and  therefore  E=^ 

b 

If  therefore  the  space  S  be  exceedingly  small  as  compared 

*  This  is  the  form  under  which  the  power  of  the  wedge  is  applied  for  the 
expressing  of  oil. 


326  THE   SCREW. 

with  Us,  there  will  be  an  exceedingly  great  resistance  E 
overcome  by  the  impact  through  that  small  space,  however 
slight  the  impact.  From  this  fact  the  enormous  amount  of 
the  resistances  which  the  wedge,  when  struck  by  the  ham- 
mer, is  made  to  overcome,  is  accounted  for.  The  power  of 
thus  subduing  enormous  resistances  by  impact  is  not  how- 
ever peculiar  to  the  wedge,  it  is  common  to  all  implements 
of  impact,  and  belongs  to  its  nature  ;  its  effects  are  rendered 
permanent  in  the  wedge  by  the  property  possessed  by  that 
implement  of  retaining  permanently  any  position  into  which 
it  is  driven  between  two  resisting  surfaces,  and  thereby  op- 
posing itself  effectually  to  the  tendency  of  those  surfaces,  by 
reason  of  their  elasticity,  to  recover  their  original  form  and 
position.  It  is  equally  true  of  any  the  slightest  direct  impact 
of  the  hammer  as  of  its  impact  applied  through  the  wedge, 
that  it  is  sufficient  to  cause  any  finite  resistance  opposed  to 
it  to  yield  through  a  certain  finite  space,  however  great  that 
resistance  may  be.  The  difference  lies  in  this,  that  the  sur- 
face yielding  through  this  exceedingly  small  but  finite  space 
under  the  blow  of  the  hammer,  immediately  recovers  itself 
after  the  blow  if  the  limits  of  elasticity  be  not  passed ; 
whereas  the  space  which  the  wedge  is,  by  such  an  impact, 
made  to  traverse,  in  the  direction  of  its  length,  becomes  a 
permanent  separation. 


THE   SCKEW. 

251.  Let  the  system  of  two  moveable  inclined  planes  re- 
presented in  fig.  p.  318.  be  formed  of  ex- 
ceedingly thin  and  pliable  laminse,  and  con- 
ceive one  of  them,  A  for  instance,  to  be 
wound  upon  a  convex  cylindrical  surface,  as 
shown  in  the  accompanying  figure,  and  the 
other,  B,  upon  a  concave  cylindrical  surface 
having  an  equal  diameter,  and  the  same  axis 
with  the  other;  then  will  the  surfaces 
EF  and  GH  of  these  planes  represent  truly 
the  threads  or  helices  of  two  screws,  one  of  them  of  the  form 
called  the  male  screw,  and  the  other  the  female  screw.  Let 
the  helix  EF  be  continued,  so  as  to  form  more  than  one  spire 
or  convolution  of  the  thread ;  if,  then,  the  cylinder  which 
carries  this  helix  be  made  to  revolve  upon  its  axis  by  the 
action  of  a  pressure  Pj  applied  to  its  circumference,  and  the 
cylinder  which  carries  the  helix  GH  be  prevented  from  re- 


THE    SCKEW.  327 

volving  upon  its  axis  by  the  opposition  of  an  obstacle  D, 
which  leaves  that  cylinder  nevertheless  free  to  move  in  a 
direction  parallel  to  its  axis,  it  is  evident  that  the  helix  EF 
will  be  made  to  slide  beneath  GH,  and  the  cylinder  which 
carries  the  latter  helix  to  traverse  longitudinally  ;  moreover, 
that  the  conditions  of  this  mutual  action  of  the  helical  sur- 
faces EF  and  GrH  will  be  precisely  analogous  to  those  of  the 
surfaces  of  contact  of  the  two  moveable  inclined  planes  dis- 
cussed in  Art.  244.  So  that  the  conditions  of  the  equili- 
brium of  the  pressures  P,  and  P,  in  the  state  bordering  upon 
motion,  and  the  modulus  of  the  system,  will  be  the  same  in 
the  one  case  as  in  the  other  ;  with  this  single  exception,  that 
the  resistance  R2  of  the  mass  on  which  the  plane  A  rests  (see 
fig.  p.  318.)  is  not,  in  the  case  of  the  screw,  applied  only  to 
the  thin  edge  of  the  base  of  the  lamina  A,  but  to  the  whole 
extremity  of  the  solid  cylinder  on  which  it  is  fixed,  or  to  a 
circular  projection  from  that  extremity  serving  it  as  a  pivot. 
Now  if,  in  equation  299,  we  assume  <?2=0,  we  shall  obtain 
that  relation  of  the  pressures  Px  and  P2  in  their  state  border- 
ing upon  motion,  which  would  obtain  if  there  were  no  fric- 
tion of  the  extremity  of  the  cylinder  on  the  mass  on  which  it 
rests  ;  and  observing  that  the  pressure  P2  is  precisely  that. 
by  which  the  pivot  at  the  extremity  of  the  cylinder  is  pressed 
upon  this  mass,  arid  therefore  the  moment  (see  Art.  176, 
equation  183)  of  the  resistance  to  the  rotation  of  the  cylinder 

2 
produced  by  the  friction  of  this  pivot  by  -P2p  tan.  <p2,  where 

p  represents  the  radius  of  the  pivot  ;  observing,  moreover, 
that  the  pressure  which  must  be  applied  at  the  circumfe- 
rence of  the  cylinder  to  overcome  this  resistance,  above  that 
which  would  be  required  to  give  motion  to  the  screw  if  there 

were  no  such  friction,  is  represented  by-  P2  -tan.  <p2,  r  being 

taken  to  represent  the  radius  of  the  cylinder,  we  obtain  for 
the  entire  value  of  the  pressure  Px  in  the  state  bordering 
upon  motion 


sin.  (*  +  ?,)  cos.  9.     2     p 

+rt< 


The  pressure  Px  has  here  been  supposed  to  be  applied  to 
turn  the  screw  at  its  circumference  ;  it  is  customary,  however, 
to  apply  it  at  some  distance  from  its  circumference  by  the 
intervention  of  an  arm.  If  a  represent  the  length  of  such  an 


328 


THE   SCREW. 


arm,  measuring  from  the  axis  of  the  cylinder,  it  is  evident 
that  the  pressure  P!  applied  to  the  extremity  of  that  arm, 
would  produce  at  the  circumference  of  the  cylinder  a  pressure 

represented  by  P^,  which  expression  being  substituted  for 

P!  "in  the  preceding  equation,  and  that  equation  solved  in 
respect  to  P1?  we  obtain  finally  for  the  relation  between  P, 
and  P3  in  their  state  bordering  upon  motion, 

m  j  sin.  (,+,.)  COB  ,     8/j.V  tan_      ) 

\af   (  cos.  (i  +  Vi+Vi)      3\r/ 


If  in  like  manner  we  assume  in  the  modulus  (equation  300) 
92=0,  and  thus  determine  a  relation  between  the  work  done 
at  the  driving  point  and  that  yielded  at  the  working  point, 
on  the  supposition  that  no  work  is  expended  on  the  friction 
of  the  pivot  ;  and  if  to  the  value  of  U,  thus  obtained  we  add 
the  work  expended  upon  the  resistance  of  the  pivot  which  is 
shown  (equation  184)  to  be  represented  at  each  revolution 
4. 

by   ^pP2  tan.  9,,  and    therefore    during  n   revolutions  by 
o 

4. 

-*^pPa,  we  shall  obtain  the  following  general  expression  for 

o 

the  modulus  ;  the  whole  expenditure  of  work   due   to  the 
prejudicial  resistances  being  taken  into  account. 


TT      TT         sn.  i  +  9    cos.  9,  ^ 

U'=U-  '  +     ^  tan' 


Representing  by  X  the  common  distance  between  the  threads 
of  the  screw,  i.  e.  the  space  which  the  nut  B  is  made  to 
traverse  at  each  revolution  of  the  screw  ;  and  observ- 

4  4  IT 

that  n^P  2=U2,  so  that  ^wP3  tan.  9,=-*—  2p  tan.  93= 

O  O      A. 


*r  P    TT  ,  .  ,  .        * 

-—  -.-.U2   tan.   9,  in  which   expression  —  =  cot.  t,   we 
6  A.     r  ** 

obtain  finally  for  the  modulus  of  the  screw 


i  sin,  fr  +  9,)  cos.  93     2  P_  J         ^         ( 

8  I    cos.  ^  +  91-f-93)       3r  } 

It  is  evidently  immaterial  to  the  result  at  what  distance 
from  the  axis  the  obstacle  D  is  opposed  to  the  revolution  of 


APPLICATIONS   OF   THE   SCKEW. 


329 


that  cylinder  which  carries  the  lamina  B  ;  since  the  amount 
of  that  resistance  does  not  enter  into  the  result  as  expressed 
in  the  above  formula,  but  only  its  direction  determined  by 
the  angle  98,  which  angle  depends  upon  the  nature  of  the 
resisting  surfaces,  and  not  upon  the  position  of  the  resisting 
point. 


APPLICATIONS  OF  THE  SCREW. 

252.  'The  accompanying  figure  represents  an  application 
of  the  screw  under  the  circumstances  described  in  the  pre- 
ceding article,  to  the  well  known  machine  called  the  V  ICE. 


»AB  is  a  solid  cylinder  carrying  on  its  surface  the  thread  of  a 
male  screw,  and  within  the  piece  CD  is  a  hollow  cylindrical 
surface,  carrying  the  corresponding  thread  of  a  female 
screw;  this  female  screw  is  prevented  from  revolving  with 
the  male  screw  by  a  groove  in  the  piece  CD,  which  carries 
it,  and  which  is  received  into  a  corresponding  projection  EF 
of  the  solid  frame  of  the  machine,  serving  it  as  a  guide ; 
which  guide  nevertheless  allows  a  longitudinal  motion  to 
the  piece  CD.  A  projection  from  the  frame  of  the  instru- 
ment at  B,  met  by  a  pivot  at  the  extremity  of  the  male 
screw,  opposes  itself  to  the  tendency  of  that  screw  to  tra- 
verse in  the  direction  of  its  length.  The  pressure  P2  to  be 
overcome  is  applied  between  the  jaws  H  and  K  of  the  vice, 
and  the  driving  pressure  Pj  to  an  arm  which  carries  round 
with  it  the  screw  AB. 

It  is  evident  that,  in  the  state  bordering  upon  motion,  the 
resistance  R  upon  the  pivot  at  the  extremity  B  of  the  screw 
AB,  resolved  in  a  direction  parallel  to  the  length  of  that 
screw,  must  be  equal  to  the  pressure  P2  (see  Art.  16.) ;  so 
that  if  we  imagine  the  piece  CD  to  become  fixed,  and  the 


330 


APPLICATIONS    OF   THE    6CKEW. 


piece  BM  to  become  moveable,  being  prevented  from  revolv 
ing,  as  CD  was,  by  the  intervention  of  a  groove  and  guide, 
then  might  the  instrument  be  applied  to  overcome  any  given 
resistance  E  opposed  to  the  motion  of  this  piece  CD  by  the 
constant  pressure  of  its  pivot  upon  that  piece. 

The  screw  is  applied  under  these  circumstances  in  the 
common  screw  press.  The  piece 
A,  fixed  to  the  solid  frame  of  the 
machine,  contains  a  female  screw 
whose  thread  corresponds  to  that 
of  the  male  screw ;  this  screw, 
when  made  to  turn  by  means  of  a 
handle  fixed  across  it,  presses  by 
the  intervention  of  a  pivot  B,  at  its 
extremity,  upon  the  surface  of  a 
solid  piece  EF  moveable  verti- 
cally, but  prevented  from  turning 
with  the  screw  by  grooves  receiv- 
ing two  vertical  pieces,  which 
serve  it  as  guides,  and  form  parts 
of  the  frame  of  the  machine. 

The  formulae  determined  in 
Art.  251.  for 'the  preceding  case 
of  the  application  of  the  screw,  obtain  ialso  in  this  case,  if 
we  assume  93— 0.  The  loss  of  power  due  to  the  friction  of 
the  piece  EF  upon  its  guides  will,  however,  in  this  calcu- 
lation, be  neglected ;  that  expenditure  is  in  all  cases  exceed- 
ingly small,  the  pressure  upon  the  guides,  whence  their 
friction  results,  being  itself  but  the  result  of  the  friction  of 
the  pivot  B  upon  its  bearings ;  and  the  former  friction  being 
therefore,  in  all  cases,  a  quantity  of  two  dimensions  in 
respect  to  the  coefficient  of  friction. 

If,  instead  of  the  lamina  A  (p.  326.)  being  fixed  upon  the 
convex  surface  of  a  solid  cylinder,  and  B  upon  the  concave 
surface  of  a  hollow  cylinder,  the  order  be  reversed,  A  being 
fixed  upon  the  hollow  and  B  on  the  solid  cylinder,  it  is  evi- 
dent that  the  conditions  of  the  equilibrium  will  remain  the 
same,  the  male  instead  of  the  female  screw  being  in  this  case 
made  to  progress  in  the  direction  of  its  length.  If,  however, 
the  longitudinal '  motion  of  the  male  screw  B  (p.  326.)  be, 
under  these  circumstances,  arrested,  and  that  screw  thus 
become  fixed,  whilst  the  obstacle  opposed  to  the  longitudinal 
motion  of  the  female  screw  A  is  removed,  and  that  screw 
thus  becomes  free  to  revolve  upon  the  male  screw,  and  also 
to  traverse  it  longitudinally,  except  in  as  far  as  the  latter 


THE   DIFFERENTIAL    SCREW. 


331 


motion  is  opposed  by  a  certain  resistance 
R,  which  the  screw  is  intended,  tinder 
these  circumstances,  to  overcome ;  then 
will  the  combination  assume  the  well 
known  form  of  the  screw  and  nut. 

To  adapt  the  formulae  of  Art.  251.  to 

this  case,  93  must  be  made  =  0,  and 
instead  of  assuming  the  friction  upon  the  extremity,  of  the 
screw  (equation  311)  to  be  that  of  a  solid  pivot,  we  must 
consider  it'  as  that  of  a  hollow  pivot,  applying  to  it  (by 
exactly  the  same  process  as  in  Art.  (251.),  the  formulas  of 
Art.  (177.)  instead  of  Art.  (176.). 


THE  DIFFERENTIAL  SCREW. 

253.  In  the  combination  of  three  inclined  planes  discussed 
in  Art.  245.,  let  the  plane  B  be  conceived  of  much  greater 
width  than  is  given  to  it  in  the  figure  (p.  319.),  and  let  it 
then  be  conceived  to  be  wrapped  upon  a  convex  cylindrical 
surface.  Its  two  edges  ab  and  cd  will  thus  become  the 
helices  of  two  screws,  having  their  threads  of  different  incli- 
nations wound  round  different  portions  of  the  same  cylinder, 


mm* 


as  represented  in  the  accompanying  figure,  where  the  thread 
of  one  screw  is  seen  winding  upon  the  surface  of  a  solid 
cylinder  from  A  to  C,  and  the  thread  of  another,  having  a 
different  inclination,  from  D  to  B. 

Let,  moreover,  the  planes  A  and  0  (p.  319.)  be  imagined 
to  be  wrapped  round  two  hollow  cylindrical  surfaces,  of 
equal  diameters  with  the  above-mentioned  solid  cylinder, 
and  contained  within  the  solid  pieces  E  and  F,  through 
which  hollow  cylinders  AB  passes.  Two  female  screws  will 
thus  be  generated  within  the  pieces  E  and  F,  the  helix  of 


332 


HUNTER'S  SCREW. 


the  one  adapting  itself  to  that  of  the  male  screw  extending 
from  A  to  0,  and  the  helix  of  the  other  to  that  upon  the 
male  screw  extending  from  D  to  B.  If,  then,  the  piece  E 
be  conceived  to  be  fixed,  and  the  piece  F  moveable  in  the 
direction  of  the  length  of  the  screw,  bnt  prevented  from 
turning  with  it  by  the  intervention  of  a  guide,  and  if  a  pres- 
sure Pj  be  applied  at  A  to  turn  the  screw  AB,  the  action  of 
this  combination  will  be  precisely  analogous  to  that  of  the 
system  of  inclined  planes  discussed  in  Art.  245.,.  and  the 
conditions  of  the  equilibrium  precisely  the  same  ;  so  that  the 
relation  between  the  pressure  Pa  applied  to  turn  the  screw 
(when  estimated  at  the  circumference  of  the  thread)  and  that 
P2,  which  it  may  be  made  to  overcome,  are  determined  by 
equation  (301),  and  its  modulus  by  equation  (302). 

The  invention  of  the  differential  screw  has  been  claimed 
by  M.  Prony,  and  by  Mr.  White  of  Manchester.  A  com- 
paratively small  pressure  may  be  made  by  means  of  it  to 
yield  a  pressure  enormously  greater  in  magnitude.*  It 
admits  of  numerous  applications,  and,  among  the  rest,  of 
that  suggested  in  the  preceding  engraving. 


HUNTER'S  SCREW. 


254.  If  we  conceive  the  plane  B  (p.  319.)  to  be  divided 
by  a  horizontal  line,  and  the  upper  part 
to  be  wrapped  upon  the  inner  or  concave 
surface  of  a  hollow  cylinder,  whilst  the 
lower  part  is  wrapped  upon  the  outer  or 
convex  circumference  of  the  same  cylin- 
der, thus  generating  the  thread  of  a  fe- 
male screw  within  the  cylinder,  and  a 
male  screw  without  it ;  and  if  the  plane 
C  be  then  wrapped  upon  the  convex  sur- 
face of  a  solid  cylinder  just  fitting  the  in- 


side  or  concave  surface  of  the  above-mentioned  hollow  cylin- 

*  It  will  be  seen  by  reference  to  equation  (301),  that  the  working  pressure 
P2  depends  for  its  amount,  not  upon  the  actual  inclinations  ii  i%  of  the  threads, 
but  on  the  diiference  of  their  inclinations ;  so  that  its  amount  may  be  enor- 
mously increased  by  making  the  threads  nearly  of  the  same  inclination.  Thus, 

neglecting  friction,  we  have,  by  equation  (301),  P2=Pi — 7—7 —r-  ;    which 

sin.  (ii  —  h) 
expression  becomes  exceedingly  great  when  ^  nearly  equals  £3. 


VARIABLE   INCLINATION   OF   THE   THREAD.  333 

der,  and  the  plane  A  upon  a  concave  cylindrical  surface  just 
capable  of  receiving  and  adapting  itself  to  the  outside  or 
convex  surface  of  that  cylinder,  the  male  screw  thus  genera- 
ted adapting  itself  to  the  thread  of  the  screw  within  the  hol- 
low cylinder,  and  the  female  screw  to  the  thread  of  that 
without  it ;  if,  moreover,  the  female  screw  last  mentioned 
be  fixed,  and  the  solid  male  screw  be  free  to  traverse  in  the 
direction  of  its  length,  but  be  prevented  turning  upon  its 
axis  by  the  intervention  of  a  guide  ;  if,  lastly,  a  moving  pres- 
sure or  power  be  applied  to  turn  the  hollow  screw,  and  a  re- 
sistance be  opposed  to  the  longitudinal  motion  of  the  solid 
screw  which  is  received  into  it ;  then  the  combination  will 
be  obtained,  which  is  represented  in  the  preceding  engraving, 
and  which  is  well  known  as  Mr.  Hunter  s  screw,  having  been 
first  described  by  that  gentleman  in  the  seventeenth  volume 
of  the  Philosophical  Transactions. 

The  theory  of  this  screw  is  identical  with  that  of  the  pre- 
ceding, the  relation  of  its  driving  and  working  pressures  is 
determined  by  equation  (301),  and  its  modulus  by  equation 
(302). 


THE  THEORY  OF  THE  SCREW  WITH  A  SQUARE  THREAD  IN  RE- 
FERENCE TO  THE  VARIABLE  INCLINATION  OF  THE  THREAD  AT 
DIFFERENT  DISTANCES  FROM  THE  AxiS. 

255.  In  the  preceding  investigation,  the  inclined  plane 
which,  being  wound  upon  the  cylinder,  generates  the  thread 
of  the  screw,  has  been  imagined  to  be  an  exceedingly  thin 
sheet,  on  which  hypothesis  every  point  in  the  thread  may  be 
conceived  to  be  situated  at  the  same  distance  from  the  axis 
of  the  screw  ;  and  it  is  on  this  supposition  that  the  relation 
between  the  driving  and  working  pressure  in  the  screw  and 
its  modulus  have  been  determined. 


Let  us  now  consider  the  actual  case  in  which  the  thread 


334:  VARIABLE   INCLINATION    OF   THE   THREAD. 

of  the  screw  is  of  finite  thickness,  and  different  elements  of 
it  situated  at  different  distance  from  its  axis. 

Let  mb  represent  a  portion  of  the  square  thread  of  a  screw, 
in  which  form  of  thread  a  line  be,  drawn  from  any  point  5  on 
the  outer  edge  of  the  thread  perpendicular  to  the  axis  ef, 
touches  the  thread  throughout  its  whole  depth  ~bd.  Let  AC 
represent  a  plane  perpendicular  to  its  axis,  and  of  the  pro- 
jection of  1)6  upon  this  plane.  Take^?  any  point  in  bd,  and 
let  q  be  the  projection  of  p.  Let  ep=r,  mean  radius  of 
thread  =R,  inclination  of  that  helix  of  the  thread  whose 
radius  is  E*=I,  inclination  of  the  helix  passing  through  p=t, 
whole  depth  of  thread  =2D,  distance  between  threads  (or 
pitch)  of  screw  =L.  Now,  since  the  helix  passing  through 
p  may  be  considered  to  be  generated  by  the  enwrapping  of 
an  inclined  plane  whose  inclination  is  i  upon  a  cylinder 
whose  radius  is  r,  the  base  of  which  inclined  plane  will  then 
become  the  arc  tq,  we  h&VB<j?£=zfo .  tan.  i.  But,  if  the  angle 
Afa  be  increased  to  2*,  pq  will  become  equal  to  the  com- 
mon distance  L  between  the  threads  of  the  screw,  and  tq 
will  become  a  complete  circle,  whose  radius  is  r  ;  therefore 
L=2*r  tan.  t,  and  this  being  true  for  all  values  of  r,  there- 
fore L= 2tfR  tan.  I.  Equating  the  second  members  of  these 
equations,  and  solving  in  respect  to  tan.  *, 

R  tan.  I  f     ON 

tan.  i= (313). 

From  which  expression  it  appears,  that  the  inclination  of  the 
thread  of  a  square  screw  increases  rapidly  as  we  recede  from 
its  edg^e  and  approach  its  axis,  and  would  become  a  right 
angle  if  the  thread  penetrated  as  far  as  the  axis.  Consider- 
ing, therefore,  the  thread  of  the  screw  as  made  up  of  an  in- 
finite number  of  helices,  the  modulus  of  each  one  of  which 
is  determined  by  equation  (312),  in  terms  of  its  correspond- 
ing inclination  £,  it  becomes  a  question  of  much  practical  im- 
portance to  determine,  if  the  screw  act  upon  the  resistance 
at  one  point  only  of  its  thread,  at  what  distance  from  its  axis 
that  point  should  be  situated,  and  if  its  pressure  be.  applied 
at  all  the  different  points  of  the  depth  of  its  thread,  as  is 
commonly  the  case,  to  determine  how  far  the  conditions  of 
its  action  are  influenced  by  the  different  inclinations  of  the 
thread  at  these  different  depths. 

*  This  may  be  called  the  mean  helix  of  the  thread.  The  term  helix  is  here 
taken  to  represent  any  spiral  line  drawn  upon  the  surface  of  the  thread ;  the 
distance  of  every  point  in  which,  from  the  axis  of  the  screw,  is  the  same. 


VARIABLE   INCLINATION    OF   THE   THREAD.  335 

We  shall  omit  the  discussion  of  the  former  case,  and  pro- 
ceed to  the  latter. 

Let  Pa  represent  the  pressure  parallel  to  its  axis  which  is 
to  be  overcome  by  the  action  of  the  screw.  ~Now  it  is  evi- 
dent that  the  pressure  thus  produced  upon  the  thread  of  the 
screw  is  the  same  as  though  the  whole  central  portion  of  it 
within  the  thread  were  removed,  or  as  though  the  whole 
pressure  P2  were  applied  to  a  ring  whose  thickness  is  As  or 
2D.  'Now  the  area  of  this  ring  is  represented  by  *  |(R+D)a 
—  (R—  D)25  ,  or  by  4<KD.  So  that  the  pressure  of  Pa,  upon 

p 
every  square  unit  of  it,  is  represented  by       -p*         Let  &r 

represent  the  exceedingly  small  thickness  of  such  a  ring 
whose  radius  is  r,  and  which  may  therefore  be  conceived  to 
represent  the  termination  of  the  exceedingly  thin  cylindrical 
surface  passing  through  the  point  p  ;  the  area  of  this  ring  is 
then  represented  by  far^r,  and  therefore  the  pressure  upon 


it  by    V,-p-pv  •>  or  by     '    ..      Now  this  is   evidently  the 

pressure  sustained  by  that  elementary  portion  of  the  thread 
which  passes  through  p,  whose  thickness  is  A/1,  and  which 
may  be  conceived  to  be  generated  by  the  enwrapping  of  a 
thin  plane,  whose  inclination  is  *,  upon  a  cylinder  whose  ra- 
dius is  r  ;  whence  it  follows  (by  equation  311)  that  the  ele- 
mentary pressure  AP15  which  must  be  applied  to  the  arm  of 
the  screw  to  overcome  this  portion  of  the  resistance  P2,  thus 
applied  parallel  to  the  axis  upon  an  element  of  the  thread, 
is  represented  by 


AP  _      .  r  \  f  sin.  Q  +  9Q  cos.  93     »P  t          \  . 

r~  (   cos..+9+9    ~~Va  9*  i  ' 


cos.(.+91+9.) 

whence,  passing  to  the  limit  and  integrating,  we  have 
R+D 


sn. 

'coS 

R-D 


Now 

sin.  0  +  9,)  cos.  93  tan.  »-f  tan.  9, 

cos.  ('  +  9,+9s)       1 — tan.  9.  tan.  9, — tan.  »  (tan.  9, 4- tan.  $} 

\  '  1      •  T«^ 

tan,  t+tan.  9t )  ___ 

~(1— tan.  9,  tan.  9,)  {I—tan.  «tan.  (9^93)  j  "  :  an*  Pi+ra11-  ' 


836  THE   SCREW    WITH   A   TRIANGULAR   THREAD. 

+tan.  fo-f  <p,)  tan.  *t.    Neglecting  dimensions  of  tan.  91  and 
tan.  <p8  above  the  first*, 

R+D 
tan.  <pj  +  tan.  i + tan.  (<pj  -j-  98)  tan.  "A/"9  -}- 

R— D 

fp^  tan.  y^dr (314). 

Substituting  in  this  expression  for  tan.  i  its  value  (equation 
313),  it  becomes 
R+D 

'a  tan.  9, + Rr  tan.  I + Ra  tan.3 1  tan.  (<pj + 9,)  + 
R-D 


Integrating  and  reducing, 


tan.  9I  tan.  fo+cp,)  j  .....  (315); 
whence  we  obtain  by  (equation  121)  for  the  modulus, 


.....  (316). 

256.  "Whence  it  follows  that  the  best  inclination  of  the 
thread,  in  respect  to  the  economv  of  power  in  the  use  of 
the  square  screw,  is  that  which  satisfies  the  equation 


The  inclination  of  thread  of  a  square  screw  rarely  exceeds 
7°,  so  that  the  term  tan.  2I  tan.  (ft  •$•$,)  rarely  exceeds  *015 
tan.  (pi+pj,),  and  may  therefore  be  neglected,  as  compared 


*  The  integration  is  readily  effected  without  this  omission  ;  and  it  might  be 
desirable  so  to  effect  it  where  the  theory  of  wooden  screws  is  under  discussion, 
the  limiting  angle  of  resistance  being,  in  respect  to  such  screws,  considerable. 
The  length  and  complication  of  the  resulting  expression  has  caused  the  omis- 
sion of  it  in  the  text. 


THE   BEAM   OF   THE    STEAM   ENGINE. 


337 


with  the  other  terms  of  the  expression;   as  also  may  the 

term  i(—  j    tan.  915  since  the  depth  2D  of  a  square  screw 

\Iv 

being  usually  made  equal  to  about  £th  of  the  diameter,  this 
term  does  not  commonly  exceed  T^  tan.  <pa. 

Omitting  these  terms,  observing  that  L=2tfR  tan.  I,  and 
eliminating  tan.  I, 


.....  (317). 
.....  (318). 


THE  BEAM  OF  THE  STEAM 

257.  Let  P,,  P,,  P3,  P4  represent  the  pressuo-es  applied  by 
the  piston  rod,  the  crank  rod,  the  air  pump-  rod,  and  the  cold 


water  pump  rod,  to  the  beam  of  a  steam  engine ;  and  sup- 
pose the  directions  of  all  these  pressures  to  be  vertical.* 

Let  the  rods,  by  which  the  pressures  P15  P2,  P3,  P4  are 
applied  to  the  beam,  be  moveable  upon  solid  axes  or  gud- 
geons, whose  centres  are  #,  d,  ~b,  6,  situated  in  the  same 
straight  line  passing  through  the  centre  C  of  the  solid  axis 
of  the  beam. 

Let  pj,  p2,  p8,  p4  represent  the  radii  of  these  gudgeons,  p  the 
radius  of  the  axis  of  the  beam,  and  <p,,  <pa,  <p9,  <p4,  p  the  limit- 
ing angles  of  resistance  of  these  axes  respectively.  Then,  if 
the  beam  be  supposed  in  the  state  bordering  upon  motion 

• 

*  A  supposition  which  in  no  case  deviates  greatly  from  the  truth,  and  any 
error  in  which  may  be  neglected,  inasmuch  as  it  can  only  influence  the  results 
about  to  be  obtained  in  as  far  as  they  have  reference  to  the  friction  of  the 
beam ;  so  that  any  error  in  the  result  must  be  of  two  dimensions,  at  least,  in 
respect  to  the  coefficient  of  friction  and  the  small  angle  by  which  any  pressure 
deviates  from  a  vertical  direction. 

22 


338  THE   BEAM   OF   THE    STEAM   ENGINE. 

by  the  preponderance  of  P,,  each  gudgeon  or  axis  being 
upon  the  point  of  turning  on  its  bearings,  the  directions  of 
the  pressures  P15  P2,  P3,  P4,  R,  will  not  be  through  the  cen- 
tres of  their  corresponding  axes,  but  separated  from  them  by 
perpendicular  distances  severally  represented  by  pt  sin.  9,,  p2 
sin.  92,  p,,  sin.  <pa,  p4  sin.  <p4,  and  p  sin.  9,  which  distances,  being 
perpendicular  to  the  directions  of  the  pressures,  are  all 
measured  horizontally. 

Moreover,  it  is  evident  that  the  direction  of  the  pressure 
Pj  is  on  that  side  of  the  centre  a  of  its  axis  which  is  nearest 
to  the  centre  of  the  beam,  since  the  influence  of  the  friction 
of  the  axis  a  is  to  diminish  the  effect  of  that  pressure  to  turn 
the  beam.  And  for  a  like  reason  it  is  evident  that  the 
directions  of  the  pressures  P2,  P8,  P4  are  farther  from  the 
centre  of  the  beam  than  the  centres  of  their  several  axes, 
since  the  effect  of  the  friction  is,  in  respect  to  each  of  these 
pressures,  to  increase  the  resistance  which  it  opposes  to  the 
rotation  of  the  beam ;  moreover,  that  the  resistance  R  upon 
the  axis  of  the  beam  has  its  direction  upon  the  same  side  of 
the  centre  C  as  P15  since  it  is  equal  and  opposite  to  the 
resultant  pressure  upon  the  beam,  and  that  resultant  would, 
by  itself,  turn  the  beam  in  the  same  direction  as  ¥l  turns  it. 
Let  now  al=Ga^  #2— C<#,  aa=Cb,  a4=Ce.  Draw  the  hori- 
zontal line  of/C<7,  and  let  the  angle  aCf=&.  Let,  moreover, 
W  be  taken  to  represent  the  weight  of  the  beam,  supposed 
to  act  through  the  centre  of  its  axis.  Then  since  P1?  P2,  P3, 
P4,  W,  R  are  pressures  in  equilibrium,  we  have,  by  the 
principle  of  the  equality  of  moments,  taking  o  as  the  point 
from_which  the  moments  are  measured,  Px  .  0/'=P2  . 
P,  .<^+P4.0£+W.0C. 

Now  of=Cf—Co=a1  cos.  8— p,  sin.  (p,— p  sin.  9,  og 
00=0,2  cos-  ^  +  p2  sin.  92  +  p  sin.  9,  oh=Ch— Co=as  cos.  d  + 
P8  sin.  93— p  sin.  9,  ok=Ck  +  Co=at  cos.  d  +  p4  sin.  94  +  p  sin.  9. 


.*.  PJ&j  cos.  d — (p^in.  9,-j-psin.  9)^  = 

P2 1&2  cos.  6  -f  (p2  sin.  9a  +  p  sin.  9)}  + 

)3  sin.  93— P  sin.  9) j  +  [ .  .  .  (319). 

i  +  p  sin.  9)  j  +  "Wp  sin.  9 

Multiplying  this  equation  by  ^,  observing  that  af  repre- 
sents the  space  described  by  the  point  of  application  of  P15 
so  that  Pj^  represents  the  work  TJ1  of  Pj ;  and  similarly 
that  P2<^2^  represents  the  work  U2  of  P2,  P3&A  that  U3  of  P8, 
and  P4&/,  that  U4  of  P4,  also  that  a£  represents  the  space  S, 


THE   BEAM   OF   THE    STEAM   ENGINE. 


339 


described  by  the  extremity  of  the  piston  rod  very  nearly  ; 
we  have 


,.(320), 


which  is  the  modulus  of  the  beam. 

Its  form  will  be  greatly  simplified  if  we  assume  cos.  d=l, 
since  6  is  small,*  suppose  the  coefficient  of  friction  at  each 
•axis  to  be  the  same,  so  that  9=^=9,  =<p8=94,  and  divide  by 
the  coefficient  of  UM  omitting  terms  above  the  first  dimen- 

sion in  —  sin,  9,  &c.  ;  whence  we  obtain  by  reduction 


-(321). 


258.  The  best  position  of  the  axis  of  the  foam. 

Let  a  be  taken  to  represent  the  length  of  the  beam,  and  x 
the  distance  aC  of  the  centre  of  its  axis  from  the  extremity 
to  which  the  driving  pressure  is  applied. 


*  In  practice  the  angle  6  never  exceeds  20°,  so  that  cos.  6  never  differs  from 
unity  by  more  than  -060807.  The  error,  resulting  from  which  difference,  in 
the  friction,  estimated  as  above,  must  in  all  cases  be  inconsiderable. 


340  THE   BEAM   OF   THE    STEAM   ENGINE. 

Let  the  influence  of  the  position  of  the  axis  on  the 
economy  of  the  work  necessary  to  open  the  valves,  to  work 
the  air-pump,  and  to  overcome  the  friction  produced  by  the 
weight  of  the  axis,  be  neglected  ;  and  let  it  be  assumed  to 
be  that,  by  which  a  given  amount  of  work  U2  may  be 
yielded  per  stroke  upon  the  crank  rod,  by  the  least  possible 
amount  u  ,  of  work  done  upon  the  piston  rod.  If,  then,  in 
equation  (321),  we  assume  the  three  last  terms  of  the  second 
member  to  be  represented  by  A,  and  observe  that  al  in  that 
equation  is  represented  by  a?,  and  #a  by  a  —  x,  we  shall 
obtain 


The  best  position  of  the  axis  is  determined  by  that  value 
of  x  which  renders  this  function  a  minimum  ;  which  value 
of  x  is  represented  by  the  equation 

x=  _  -  _  .  (322.) 


If  p2>p,,  then!—  —  J  >1  and  a?<-|#;  in  this  case,  there- 

fore, the  axis  is  to  be  placed  nearer  to  the  driving  than  to 
the  working  end  of  the  beam.  If  p2<pj,  the  axis  is  to  be 
fixed  nearer  to  the  working  than  to  the  driving  end  of  the 
beam. 

259.  It  has  already  been  shown  (Art.  168.),  that  a 
machine  working,  like  the  beam  of  a  steam  engine,  under 
two  given  pressures  about  a  fixed  axis,  is  worked  with  the 
greatest  economy  of  power  when  both  these  pressures  are 
applied  on  the  same  side  of  the  axis.  This  principle  is 
manifestly  violated  in  the  beam  engine;  it  is  observed  in 
the  engine  worked  by  Crowther's  parallel  motion,*  and  in 
the  marine  engines  recently  introduced  by  Messrs.  Seaward, 
and  known  as  the  Gorgon  engines.  It  is  difficult  indeed  to 
defend  the  use  of  the  beam  on  any  other  legitimate  ground 
than  this,  that  in  some  degree  it  aids  the  fly-wheel  to 
equalise  the  revolution  of  the  crank  arm,f  an  explanation 

*  As  used  in  the  mining  districts  of  the  north  of  England. 

f  The  reader  is  referred  to  an  admirable  discussion  of  the  equalising  power 
of  the  beam,  by  M.  Coriolis,  contained  in  the  thirteenth  volume  of  the  Journal 
de  VEcole  Poly  technique. 


THE   CKANK. 


341 


which  does  not  extend  to  its  use  in  pumping  engines, 
where,  nevertheless,  it  retains  its  place;  adding  to  the 
expense  of  construction,  and,  by  its  weight,  greatly  increas- 
ing the  prejudicial  resistances  opposed  to  the  motion  of  the 
engine. 


THE  CKANK. 

260.  The  modulus  of  the  crank,  the  direction  of  the  resist- 
ance being  parallel  to  that  of  the  driving  pressures. 

Let  CD  represent  the  arm  of  the  crank,  and  AD  the  con- 
necting rod.  And  to  simplify  the 
investigation,  let  the  connecting 
rod  be  supposed  always  to  retain 
its  vertical  position.*  Suppose  the 
weight  of  the  crank  arm  CD,  act- 
ing through  its  centre  of  gravity, 
to  be  resolved  into  two  other 
weights  (Art.  16),  one  of  which  W2 
is  applied  at  the  centre  C  of  its  axis 
and  the  other  at  the  centre  c  of 
the  axis  which  unites  it  with  the 
connecting  rod.  Let  this  latter 
weight,  when  added  to  the  weight 
of  the  connecting  rod,  be  repre- 
sented by  Wj.  Let  P2  represent  a 
/  pressure  opposed  to  the  revolution 
x-;.  ^S  of  the  crank,  which  would  at  any 

instant  be  just  sufficient  to  balance 

the  driving  pressure  Pa  transmitted  through  the  connecting 
rod  ;  and  to  simplify  the  investigation,  let  us  suppose  the 
direction  of  the  pressure  P2  to  be  vertical  and  downwards. 

Let  Cc=a,  CA1=a1,  CA2=&2,  <?CW2=d,  radii  of  axes  C 
and  c=p,,  p2,  lim.  /s  of  resistance  =<p,,  <pa,  W=  whole  weight 
of  crank  arm  and  connecting  rod=W1  +  Wa. 

Since  the  crank  arm  is  in  the  state  bordering  upon 
motion,  the  perpendicular  distance  of  the  direction  of  the 
resistance  upon  its  axis  C  from  the  centre  of  that  axis,  is 


*  Any  error  resulting  from  this  hypothesis  affecting  the  conditions  of  the 
question  only  in  as  far  as  the  friction  is  concerned,  and  being  of  two  dimen- 
sions at  least  in  terms  of  the  coefficient  of  friction  and  the  small  angular  devi- 
ation of  the  connecting  rod  from  the  vertical. 


342 


THE   CKANK. 


represented  by  px  sin.  <pt  (Art.  153.).     The  resistance  is  also 
equal  to  Pa  ±  (Pa + W) ;  Px  being  supposed  greater  than  P2  +  W; 

and  the  sign  ±  being  taken 
according  as  the  direction  of 
P,  is  downwards  or  upwards, 
or  according  as  the  crank  arm 
is  describing  its  descending  or 
ascending  arc.  Whence  it 
follows,  that  the  moment  of 
the  resistance  of  the  axis  about 
its  centre  is  represented  by 
iP,±(P,  +  W)j  Pl  sin.  9, 
Now  the  pressures  P15  P2,  and 
the  resistance  of  the  axis,  are 
pressures  in  equilibrium. 
Therefore,  by  the  principle  of 
the  equality  of  moments,  ob- 
serving that  the  driving  pressure  is  represented  by  PjSfcW,, 
according  as  the  arm  is  descending  or  ascending, 

(P,±W,)  «,=PA  +  SP,±(P,  +  W)}  Pl  sin.  ?, 

Since  moreover  the  axis  0,  which  unites  the  connecting 
rod  and  the  crank  arm,  is  upon  the  point  of  turning  upon 
its  bearings,  the  direction  of  the  pressure  Px  is  not  through 
the  centre  of  that  axis,  but  distant  from  it  by  a  quantity 
represented  by  p2  sin.  <pa,  which  distance  is  to  be  measured 
on  that  side  of  the  centre  o  which  is  nearest  to  C,  since  the 
friction  diminishes  the  effect  of  Px  to  turn  the  crank  arm. 


!= a  sin.  d—  p3  sin.  <p2 


(323). 


Substituting  this  value  of  #,  in  the  preceding  equation, 
O  (a  sin.  0-p2  sin.  9,)=I>,+  { 


(324). 


Pl  sn.  <pa 
Transposing  and  reducing 

P,{a  sin.  &—  pa  sin.  ?2—  p,  sin.  9,}  =P2K±p1  sin.  9,}  ± 
"Wp1  sin.  91=fW1(#  sin.  $—  pa  sin.  9,); 

which  is  the  relation  between  Pt  and  P2  in  their  state  bor- 
dering upon  motion.  JSTow  if  Ad  represent  an  exceedingly 
small  angle  described  by  the  crank  arm,  a^b  will  represent 
the  space  through  which  the  resistance  P2  is  overcome 
whilst  that  angle  is  described,  and  P2#2A0  will  represent  the 


THE   CRANK.  343 

increment  AtT2  of  the  work  yielded  by  the  crank  whilst  that 
small  angle  is  described.     Multiplying  the  above  equation 
by  a^y  we  have 
P^Ja  sin.  A—  p2  sin.  <p2—  p,  sin.  ^l\A&={a9±pl  sin.  91}A"D'3± 

"Wa2px  sin.  ^AdqpW^,,  (a  sm-  &—  P*  sin-  ^)A^  •  •  •  •  (325)- 
whence  passing  to  the  limit,  integrating  from  0=6  to  d= 
tf—  0,  and  dividing  by  #2 

P,  |2acos.6—  (*—  28) 

W  <y_26)  Pi  sin.  ^qp  Wj  {2a  cos.  0—  Pa  (V—  26)  sin.  <p2}  .  .  (326). 

Now,  let  it  be  observed  that  2a  cos.  6  represents  the  pro- 
jection of  the  path  of  the  point  c  upon  the  vertical  direction 
of  P,,  whilst  the  arm  revolves  between  the  positions  6  and 
tf  _6  ;  so  that  P^a  cos.  6  represents  (Art.  52.)  the  work 
Uj  done  by  P,  upon  the  crank  whilst  the  arm  passes  from 
one  of  these  positions  to  the  other,  or  whilst  the  work  U2  is 

TJ 
yielded  by  the  crank.     Whence  it  follows  that  PI=^J  f  sec.  6. 

Substituting  this  value  of  P1?  and  reducing  we  obtain 


±     sin.  9,     U2  ±W  (-rr-26)  Pl  sin.  V^W,  [2a  cos.  0  — 
P2  (*—  20)  sin.  9a{  .....  (327). 

By  which  equation  is  determined  the  modulus  of  the  crank 
in  respect  to  the  descending  or  ascending  stroke,  according 
as  we  take  the  upper  or  lower  signs  of  the  ambiguous  terms. 
Adding  these  two  values  of  the  modulus  together,  and 
representing  by  Uj  the  whole  work  of  P,,  and  by  TJ2  the 
whole  work  of  P2,  whilst  the  crank  arm  makes  a  complete 
revolution,  also  by  ut  the  work  of  P2  in  the  down  stroke, 
and  i£2  in  the  up  stroke,  we  obtain 


-  -0    e  sin-     sin-  ?>  =v, 


K-^2)sin.  9,  .....  (328), 
which  is  the  modulus  of  the  crank  in  respect  to  a  vertical 


344  THE   CRANK. 

direction  of  the  driving  pressure  and  of  the  resistance,  the 
arm  being  supposed  in  each  half  revolution,  first,  to  receive 
the  action  of  the  driving  pressure  when  at  an  inclination  of 
©  to  the  vertical,  and  to  yield  it  when  it  has  again  attained 
the  same  inclination,  so  as  to  revolve  under  the  action  of 
the  driving  pressure  through  the  angle  if  —  2©. 

In  the  double-acting  engine,  u^—  u^=0  ;  in  the  single-act- 
ing engine  ^=0.  The  work  expended  by  reason  of  the 
friction  of  the  crank  is  therefore  less  in  the  latter  engine  than 
in  the  former,  when  the  resistance  P2  is  applied,  as  shown 
•in  the  figure,  on  the  side  of  the  ascending  arc. 

If  the  arm  sustain  the  action  of  the  driving  pressure  con- 
stantly, 9=0,  and  the  modulus  becomes,  for  the  double-act- 
ing engine, 


or,  dividing  by  the  co-efficient  of  U,  and  neglecting  dimen- 
sions above  the  first  in  sin.  91?  sin.  <p2, 


The  modulus  not  involving  the  symbol  W  which  repre- 
sents the  weight  of  the  crank,  it  is  evident  that  so  long  as  Pt 
and  P2  are  vertical  and  P,  greater  than  P2  +  "W,  the  economy 
of  power  in  the  use  of  the  crank  is  not  at  all  influenced  by 
its  weight  and  that  of  the  connecting  rod,  the  friction  being 
upon  the  whole  as  much  diminished  by  reason  of  that  weight 
in  the  ascending  stroke  as  it  is  increased  by  it  in  the  descend- 
ing stroke. 

It  is  evident,  moreover,  that  if  the  friction  produced  by 
the  weight  of  the  crank  be  neglected,  the  modulus  above  de- 
duced, for  the  case  in  which  the  directions  of  the  pressures 
P,  and  P3  are  vertical,  applies  to  every  case  in  which  the 
directions  of  those  pressures  are  parallel. 

The  condition  P,>P2-f-W  evidently  obtains  in  every  other 
position  of  the  crank  arm,  if  it  obtain  in  the  horizontal  position. 

Now,  in  this  position,  P2=—  P15  if  we  neglect  friction.     The 

ai 

required  condition  obtains,  therefore,  if  ?,>—?!  -f-W.  To 

a* 

satisfy  this  condition,   #2  must  be  greater  than  #,  or  the 

resistance  be  applied  at  a  perpendicular  distance  from  the 


THE   DEAD   POINT   IN   THE   CRANK.  345 

axis  greater  than  the  length  of  crank  arm,  and  so  much 
greater,  that  Pl   (1  --  j  may  exceed  "W.     These  conditions 

\  #2/ 

commonly  obtain  in  the  practical  application  of  the  crank. 

261.  Should  it,  however,  be  required  to  determine  the  mo- 
dulus in  the  case  in  which  Px  is  not,  in  every  position  of  the 
arm,  greater  than  P,-f  W,  let  it  be  observed,  that  this  condi- 
tion does  not  affect  the  determination  of  the  modulus  (equa- 
tion 327)  in  respect  to  the  descending,  but  only  the  ascend- 
ing stroke  ;  there  being  a  certain  position  of  the  arm  as  it 
ascends  in  which  the  resultant  pressure  upon  the  axis  repre- 
sented by  the  formula  {P,—  (P2+W)j  ,  passing  through  zero, 
is  afterwards  represented  by  |(P2-f-  W)—  P^  ;  and  when  the 
arm  has  still  further  ascended  so  as  to  be  again  inclined  to  the 
vertical  at  the  same  angle,  passes  again  through  zero,  and  is 
again  represented  by  the  same  formula  as  before.  The  value 
of  this  angle  may  be  determined  by  substituting  Pa  for 
Pa  +  W  inequation  (324),  and  solving  that  equation  in  re- 
spect to  A.  Let  it  be  represented  by  ^  ;  let  equation  (325) 
be  integrated  in  respect  to  the  ascending  stroke  from  0=0 
to  0==^,  the  work  of  P2  through  this  angle  being  represented 
by  ul  ;  let  the  signs  of  all  the  terms  involving  px  sin.  <pj  then 
be  changed,  which  is  equivalent  to  changing  the  formula  re- 
presenting the  pressure  upon  the  axis  from  {Px—  (P2+W)£ 
to  {(P2+  W)—  Pa};  and  let  the  equation  then  be  integrated 

from  6=6l  to  0=5,  the  work  of  Pa  through  this  angle  being  re- 


presented  by  u^\  2(1^  +  wa)  will  then  represent  the  whole 
work  U2  done  by  P2  in  the  ascending  arc.  To  determine 
this  sum,  divide  the  first  integral  by  the  co-efficient  of  ul9 
and  the  second  by  that  of  u9,  add  the  resulting  equations, 
and  multiply  their  sum  by  2  ;  the  modulus  in  respect  to  the 
ascending  arc  will  then  be  determined  ;  and  if  it  be  added 
to  the  modulus  in  respect  to  the  descending  arc,  the  modu- 
lus in  respect  to  an  entire  revolution  will  be  known. 


THE  DEAD  POINTS  IN  THE  QRANK. 

362.  If  equation  (324)  be  solved  in  respect  to   P,  it  be- 
comes 


346  THE   DOUBLE    CIJANK. 

p  _p   (  ^±PiSJn.  9,  _  ) 

2  1  a-am.  6—  p2  sin.  <pa—  p1  sin.  ?,  f 


'Wp1  sin.^—  •'WXflBin.d—  pa  sin.  98) 
a  sin.  4—  pa  sin.  <p2—  p:  sin.  9t 

In  that  position  of  the  arm,  therefore,  in  which 
sin.O=f''sin-^  +  p-sin-'P-  ....  (330), 

the  driving  pressure  P,  necessary  to  overcome  any  given  re- 
sistance P2  opposed  to  the  revolution  of  the  crank,  assumes 
an  infinite  value.  This  position  from  which  no  finite  pres- 
sure acting  in  the  direction  of  the  length  of  the  connecting 
rod  is  sufficient  to  move  the  arm,  when  it  is  at  rest  in  that 
position,  is  called  its  dead  point. 

Since  there  are  four  values  of  6,  which  satisfy  equation 
(330)  two  in  the  descending  and  two  in  the  ascending  semi- 
revolution  of  the  arm,  there  are,  on  the  whole,  four  dead 
points  of  the  crank.*  The  value  of  Pl  being,  however,  in  all 
cases  exceedingly  great  between  the  two  highest  and  the  two 
lowest  of  these  positions,  every  position  between  the  two 
former  and  the  two  latter,  and  for  some  distance  on  either 
side  of  these  limits,  is  practically  a  dead  point. 


THE  DOUBLE  CRANK. 

263.  To  this  crank,  when  applied  to  the  steam  engine,  are 
affixed  upon  the  same  solid  shaft,  two  arms  at  right  angles 
to  one  another,  each  of  which  sustains  the  pressure  of  the 
steam  in  a  separate  cylinder  of  the  engine,  which  pressure  is 
transmitted  to  it,  from  the  piston  rod,  by  the  intervention  of 
a  beam  and  connecting  rod  as  in  the  marine  engine,  or  a 
guide  and  connecting  rod  as  in  the  locomotive  engine. 


*  It  has  been  customary  to  reckon  theoretically  only  two  dead  points  of  the 
crank,  one  in  its  highest  and  the  other  in  its  lowest  position.  Every  practical 
man  is  acquainted  with  the  fallacy  of  this  conclusion. 


THE   DOUBLE   CKANK. 


347 


Fig.  1. 


In  either  case,  the  connecting  rods 
may  be  supposed  to  remain  con- 
stantly parallel  to  themselves,  and 
the  pressures  applied  to  them  in 
different  planes  to  act  in  the  samg 
plane,*  without  materially  affecting 
the  results  about  to  be  deduced.f 

Let  the  two  arms  of  the  crank  be 
supposed  to  be  of  the  same  length  a ; 
let  the  same  driving  pressure  Pl  be 
supposed  to  be  applied  to  each  ;  and 
let  the  same  notation  be  adopted  in 
other  respects  as  was  used  in  the 
case  of  the  crank  with  a  single  arm; 
and,  first,  let  us  consider  the  case 
represented  in  fig.  1,  in  which  both 
arms  of  the  crank  are  upon  the  same 
side  of  the  centre  C. 


Let  the  angle  W^B^d  ;    therefore  W1CE=^+^  :  whence 

it  follows  by  precisely  the  same  reasoning  as  in  Art.  260., 
that  the  perpendicular  upon  the  direction  of  the  driving 
pressure  applied  by  the  connecting  rod  AB  is  represented 
(see  equation  323)  by  a  sin.  6—  p2  sin.  <pa,  and  the  per- 
pendicular upon  the  pressure  applied  by  the  rod  ED  by 


a 


cos-  &—  P2  sin.  9,.      Let  now 


be  taken  to  represent  the  perpendicular  distance  from  the 
axis  C,  at  which  a  single  pressure,  equal  to  2P,,  must  be  ap- 
plied, so  as  to  produce  the  same  effect  to  turn  the  crank  as 
is  produced  by  the  two  pressures  actually  applied  to  it  by 
the  two  connecting  rods  ;  then,  by  the  principle  of  the  equa- 
lity of  moments, 


2P1«1=P1(«  sin.  &—  pQ  sin.  9a)  +  P1(«  cos.  &—  pa  sin.  9,)  ; 
/.  al=^a  (sin.  6  -f-  cos.  d)—  pa  sin.  9,  ; 


*  This  principle  will  be  more  fully  discussed  by  a  reference  to  the  theory  of 
statical  couples.  (See  Pritchard  on  Statical  Couples.) 

f  The  relative  dimensions  of  the  crank  arm  and  connecting  rod  are  here  sup- 
posed to  be  those  usually  given  to  these  parts  of  the  engine ;  the  supposition 
does  not  obtain  in  the  case  of  a  short  connecting  rod. 


34:8  THE  DOUBLE   CRANK. 

a  I  .  if  *\ 

/.  #,=  —I  sin.  &  cos.  j+cos.d  sin.  -r  I  —  pa  sin.  <pa= 


which  expression  becomes  identical  with  the  value  of  #n  de- 
termined by  equation  (323),  if  in  the  latter  equation  a  be 

replaced  by  -fL,  and  6  by  6  +-.    "Whence  it  follows  that  the 

4/2      .  * 

conditions  of  the  equilibrium  of  the  double  crank  in  the 
state  bordering  upon  motion,  and  therefore  the  form  of  the 
modulus,  are,  whilst  both  arms  are  on  the  same  side  of  the 
centre,  precisely  the  same  as  those  of  the  single  crank,  the 
direction  of  whose  arm  bisects  the  right  angle  BCE,  and 
the  length  of  whose  arm  equals  the  length  of  either  arm  of 
the  double  crank  divided  by  |/2. 

Now,  if  6l  be  taken  to  represent  the  inclination  WXCF  of 
this  imaginary  arm  to  "W^C,  both  arms  will  be  found  on  the 

same  side  of  the  centre,  from  that  position  in  which  6t  =  _ 
to  that  in  which  it  equals  (  *  —  -I.  If,  therefore,  we  substi- 

tute -  for  6,  in  equations  (326),  and  for  &,  —  ,  and  add  these 

4/2 

equations  together,  the  symbol  2  U2  in  the  resulting  equa- 
tion will  represent  the  whole  work  yielded  by  the  working 
pressure,  whilst  both  arms  remain  on  the  same  side  of  the 
centre,  in  the  ascending  and  the  descending  arcs.  We  thus 
obtain,  representing  the  sum  of  the  driving  pressures  upon 
the  two  arms  by  P:, 

2P>-     (pasin.<p2  +  Pl  8^.901=211,  .....  (331).* 


Throughout  the  remaining  two  quadrants  of  the  revolution 
of  the  crank,  the  directions  of  the  two  equal  and  parallel 
pressures  applied  to  it  through  the  connecting  rods  being 
opposite,  the  resultant  pressure  upon  the  axis  is  represented 
by  (P.  +  W),  instead  of  |P,±  (P2  +  W)  {  ;  whilst,  in  other 
respects,  the  conditions  of  the  equilibrium  of  the  state  bor 

*  Whewell's  Mechanics,  p.  25. 


THE   DOUBLE    CEANK. 


849 


dering  upon  motion  remain  the  same  as  before  ;  that  is,  the 
same  as  though  the  pressure  P:  were 
applied  to  an  imaginary  arm,  whose 

length  is  — - —t  and  whose  position  co- 
incides with  OF.  Now,  referring  to 
equation  (324),  it  is  apparent  that 
this  condition  will  be  satisfied  if,  in 
that  equation,  the  ambiguous  sign  of 
(P2  +  W)  be  suppressed,  and  the 
value  of  P!  in  the  second  member, 
w^hich  is  multiplied  by  p,  sin.  9a,  be 
assumed  =0 ;  by  which  assumption 
the  term  — px  sin.  9,  will  be  made  to 
disappear  from  the  left-hand  member 
of  equation  (325),  and  the  ambiguous 
signs  which  affect  the  first  and  second 
terms  of  the  right-hand  member  will  become  positive.  Now, 
these  substitutions  being  made,  and  the  equation  being  then 

integrated,  first,  between  the  limits  0  and  -,  and  then  be- 
tween the  limits  —  and  ir,  the  symbol  U3  in  it  will  evidently 

represent  the  work  done  during  each  of  those  portions  of  a 
semi-revolutiou  of  the  imaginary  arm  in  which  the  two  real 
arms  of  the  crank  are  not  on  the  same  side  of  the  centre. 
Moreover,  the  integral  of  that  equation  between  the  limits  0 

and  j?  is  evidently  the  same  with  its  integral  between  the 

limits  -j-  and  if.  Taking,  therefore,  twice  the  former  inte- 
gral, we  have 


P.  sin.  9, 


sn. 


-        sn. 


Dividing    this     equation     by    (0a  +  px  sin.  9,),     or    by     #a 

(1  H  —  -  sin.  9,  )>  and  neglecting  terms  above  the  first  dimen- 
G>9  ' 

Bion  in  sin.  9!  and  sin.  92, 


350  THE   DOUBLE    CRANK. 


-  £  sin.9l)- 
j  p2  sin.  <pa  I  ; 


in  which  equation  2U2  represents  the  work  done  in  the 
descending  or  ascending  arcs  of  the  imaginary  arm,  accord- 
ing as  the  ambiguous  sign  is  taken  positively  or  negatively. 
Taking,  therefore,  the  sum  of  the  two  values  of  the  equation 
given  by  the  ambiguous  sign,  and  representing  by  4U2  the 
whole  work  done  in  the  descending  and  ascending  arcs,  dur- 
ing those  portions  of  each  complete  revolution  when  both  of 
the  arms  are  not  on  the  same  side  of  the  centre,  we  have 


or,  observing  that  cos.  7  =  ~~F> 


2P,     0(|/2-l)-X  4/2-1)      sin.  9l-      Pa  sin.  9,      = 

p1  sin.  <pj. 


Adding  this  equation  to  equation  (331),  and  representing  by 
U2  the  entire  work  yielded  during  a  complete  revolution  of 
the  imaginary  arm, 


2P,     a  4/2  -  a(|/2  -  1)     sin.  9,  -    (2Pi  sin.?,  +^ 


But  if  U1  represent  the  whole  work  done  by  the  driving 
pressures  at  each  revolution  of  the  imaginary  arm,  then 

4—  P1=U1.     Since  2  —p  is  the  projection  of  the  space 


THE   CRANK   GUIDE.  351 

described  by  the  extremity  of  the  arm  during  the  ascending 
and  descending  strokes  respectively,  therefore  2Pl  =  —  j=. 
Substituting  this  value  for  2PX, 


IT      l-= 


sin.^..  ..(332), 

which  is  the  modulus  of  the  double  crank,  the  directions  of 
the  driving  pressure  and  the  resistance  being  both  supposed 
vertical;  or  if  the  friction  resulting  from  the  weight  of  the 
crank  be  neglected,  and  W  be  therefore  assumed  =0,  then 
does  the  above  equation  represent  the  modulus  of  the 
double  crank,  whatever  may  be  the  direction  of  the  driving 
pressure,  provided  that  the  direction  of  the  resistance  be 
parallel  to  it.  Dividing  by  the  coefficient  of  Uj,  and 
neglecting  terms  of  more  than  one  dimension  in  sin.  91  and 
sin.  <pf, 


^  sin.  ?,)  | 


(333). 


THE  CRANK  GUIDE. 

264.  In  some  of  the  most  important  applications  of  the 
steam  engine,  the  crank  is  made  to  receive*  its  continuous 
rotatory  motion,  from  the  alternating  rectilinear  motion  of 
the  piston  rod,  directly  through  the  connecting  rod  of  the 
crank,  without  the  intervention  of  the  beam  or  parallel 
motion ;  the  connecting  rod  being  in  this  case  jointed  at  one 
extremity,  to  the  extremity  of  the  piston  rod,  and  the  oblique 
pressure  upon  it  which  results  from  this  connexion  being 
sustained  by  the  intervention  of  a  cross  piece  fixed  upon  it, 
and  moving  between  lateral  guides.* 

*  This  contrivance  is  that  well  known  as  applied  to  the  locomotive  carriage. 


352  THE    CEANK   GUIDE. 


Let  the  length  CD  of  the  connecting  rod  be  represented 
by  J,  and  that  BD  of  the  crank  arm  by  a,  and  let  ft  and  Pa 
in  the  above  figure  be  taken  respectively,  to  represent  the 
pressure  upon  the  piston  rod  of  the  engine  and  the  connect- 
ing rod  of  the  crank,  and  RS  to  represent  the  direction  of 
the  resistance  of  the  guide  in  the  state  bordering  upon 
motion  by  the  excess  of  the  driving  pressure  Pa.  Then  is 
RS  inclined  to  a  perpendicular  to  the  direction  of  the  guides, 
or  of  the  motion  of  the  piston  rod,  at  an  angle  equal  to  the 
limiting  angle  of  resistance  (Art.  lil)  of  the  surfaces  of  con- 
tact of  the  guides. 

Since,  moreover,  P1?  P2,  R  are  pressures  in  equilibrium, 

B  P3     sin.  P,CS 

Let  /BCD =4  ;  limiting  angle  of  resistance  of  guide  =9; 
therefore,  ^08=5-9,  paCS=^+9-4; 


•-•FT     t«  „    j-a^Fi) V* 

sm.j-  —  (4— <pU 


Let  BD  =  a,  CD  =  &,  and  DBC  =  4n  and  assume  P,  to 
remain  constant,  Pj  being  made  to  vary  according  to  the 
conditions  of  the  state  bordering  upon  motion  ; 


^P,  .  AAC=—  P,  .  ABC=—  P,  .  A  («  cos.  &,+b  cos.  4)= 

P2  sec.  9  cos.  (4—9)  (a  sin.  ^A^-j-5  sin. 
AUa=-Pa(ABC)cos.  4=P2(asin.  6^ 

7T  0 

.•.UI=P,sec.9|«  /  sin.4lcos.(4—  9)^^^  /  sin.  4  cos.  (4—  9)$}. 


THE   FLY-WHEEL*  353 

<k 


U,=P3  {a J  sin.  ^  cos.  &d&1  + 1 J  sin.  &  cos. 
o  o 

The  second  integral  in  each  of  these  equations  vanishes 
between  the  prescribed  limits ;  also  sin.  &  =  T  sin.  ^ ;   there- 

a? 
fore  cos.  d  =  (1—  —  sin. 3^)*; 

7T  7T 

.*.  U,=Pa#  /  sin.  6l  cos.  ^1=P2«  /  (1  —  -TS  sin. a^,)*  sin.  ^^=s 


TJ^Pa^sec.  9  /  sin.  6l  cos.  (^—  9)^1=Pa^  /  sin.  ^  cos.  ^, 
o  o 

7T  7T 

P,a  tan.  9  /  sin.  6  sin.  ^1^1=Ua+Pa^-tan.  9  /  sin.  9^  ^= 

l 


whence  eliminating  P8  and  reducing,  we  obtain 


which  is  the  modulus  of  the  crank  guide. 

THE  FLY-WHEEL. 
265.  TA0  angular  velocity  of  the  fly-wheel. 

Let  Px  be  taken  to  represent  a  constant  pressure  applied 
through  the  connecting  rod  to  the  arm  of  the  crank  of  a 

*  Church's  Diff.  and  Int.  Gal.     Art.  199. 
23 


854 


THE    FLY-WHEEL. 


steam  engine;  suppose  the  direction  of  this  pressure  to 
remain  always  parallel  to  itself,  and  let  P2  represent  a  con- 
stant resistance  opposed  to  the  revolution  of  the  axis  which 
carries  the  fly-wheel,  by  the  useful  work  done  and  the  pre- 
judicial resistances  interposed  between  the  axis  of  the 
fly-wheel  and  the  working  points  of  the  machine. 

Let  the  angle  ACB=$,  CB=&,  CP2=«2. 

Now  the  projection,  upon  the  direction  of  P15  of  the  path 
of  its  point  of  application  B  to  the  crank  arm,  whilst  that 
arm  describes  the  angle  ACB,  is  AM,  therefore  (Art.  52.), 
the  work  done  by  Yl  upon  the  crank,  whilst  this  angle  is 
described,  is  represented  by  P,  .  AM,  or  by  P,  a  vers.  d. 
And  whilst  the  crank  arm  revolves  through  the  angle  $,  the 
resistance  P2  is  overcome  through  the  arc  of  a  circle  sub- 
tended by  the  same  angle  4,  but  whose  radius  is  «2,  or 
through  a  space  represented  by  aj.  So  that,  neglecting  the 
friction  of  the  crank  itself,  the  work  expended  upo'i  the 
resistances  opposed  to  its  motion  is  represented  by  P2#a0,  and 
the  excess  of  the  work  done  upon  it  through  the  angle  ACB 
by  the  moving  power,  over  that  expended  during  the  same 
period  upon  the  resistances,  is  represented  by 

Pavers.  4— P2^ (336). 

Now  2#Pj  represents  the  work  done  by  the  moving  pressure 
P,  during  each  effective  stroke  of  the  piston,  and  2tf<22P2  the 
work  expended  upon  the  resistance  during  each  revolution 
of  the  fly-wheel;  so  that  if  m  represent  the  number  of 
strokes  made  by  the  piston  whilst  the  fly-wheel  makes  one 


THE   FLY-WHEEL.  355 

revolution,  and  if  the  engine  be  conceived  to  have  attained 
its   state    of   uniform   or  steady   action  (Art.   146.),   then 


.•.«J>.=£*Pi  .....  (337). 

Eliminating  from  equation  (336)  the  value  of  #3P2  deter- 
mined by  this  equation,  we  obtain  for  the  excess  of  the  work 
done  by  the  power  (whilst  the  angle  6  is  described  by  the 
crank  arm),  over  that  expended  upon  the  resistance,  the 
expression 

P^jvers.  *-^j  .....  (338). 

But  this  excess  is  equal  to  the  whole  work  which  has  been 
accumulating  in  the  different  moving  parts  of  the  machine, 
whilst  the  angle  6  is  described  by  the  arm  of  the  crank  (Art. 
145).  Now,  let  the  whole  of  this  work  be  conceived  to  have 
been  accumulated  in  the  fly-wheel,  that  wheel  being  pro- 
posed to  be  constructed  of  such  dimensions  as  sufficiently  to 
equalise  the  motion,  even  if  no  work  accumulated  at  the 
same  time  in  other  portions  of  the  machinery  (see  Art.  150.), 
or  if  the  weights  of  the  other  moving  elements,  or  their 
velocities,  were  comparatively  so  small  as  to  cause  the  work 
accumulated  in  them  to  be  exceedingly  small  as  compared 
with  the  work  accumulated  during  the  same  period  in  the 
fly-wheel.  Now,  if  I  f  represent  the  moment  of  inertia  of  the 
fly-wheel,  ^  the  weight  of  a  cubic  foot  of  its  material,  a1  its 
angular  velocity  when  the  crank  arm  was  in  the  position 
CA,  and  a  its  angular  velocity  when  the  crank  arm  has 

passed  into  the  position  CB  ;  then  will  %  —  (a2—  a^)  represent 

the  work  accumulated  in  it  (Art.  75.)  between  these  two 
positions  of  the  crank  arm,  so  that 


.....  (339). 


266.  The  positions  of  greatest  and  least  angular  velocity  of 
the  fly-wheel. 

If  we  conceive  the  engine  to  have  acquired  its  state  of 
steady  or  uniform  motion,  the  aggregate  work  done  by  the 


356 


THE   FLY-WHEEL. 


power  being  equal  to  that  expended  upon  the  resistances, 
then  will  the  angular  velocity  of  the  fly-wheel  return  to  the 
same  value  whenever  the  wheel  returns  to  the  same  position  ; 
so  that  the  value  of  OLI  in  equation  (339)  is  a  constant,  and 
the  value  of  a  a  function  of  6  ;  a  assumes,  therefore,  its  mini- 
mum and  maximum  values  with  this  function  of  d,  or  it  is  a 

minimum  when  --  =  0,  and  ->0>  and  a  maximum  when 


da?      A          ,  d?a?     _ 

-^-=0,  and  w< 
therefore  -^-=0,  when 


-r,    ,  da?        . 

But  -=s 


.          m 
sin.  d— — 


.  m  ,    (Pa?  . 

_  -,  and  w=  cos.  I, 


(340.) 


Now  this  equation  is  evidently  satisfied  by  two  values  of 
d,  one  of  which  is  the  supplement  of  the  other,  so  that  if  ^ 
represent  the  one,  then  will  (*—>])  represent  the  other; 
which  two  values  of  9  give  opposite  signs  to  the  value  cos. 
6  of  the  second  differential  co-efficient  of  a2,  the  one  being 
positive  or  >0,  and  the  latter  negative  or  <0.  The  one 
value  corresponds,  therefore,  to  a  minimum  and  the  other 
•to  a  maximum  value  of  a.  If,  then,  we  take  the  angle  ACB 

Wit 

in  the  preceding   figure,  such  that  its  sine  may  equal  — 

(equation  340),  then  will  the  position  CB  of  the  crank  arm 
be  that  which  corresponds  to  the  minimum  angular  velocity 


« 


THE   FLY-WHEEL.  357 

of  the  fly-wheel ;  and  if  we  make  the  angle  ACE  equal  to 
the  supplement  of  ACB,  then  is  CE  the  position  of  the 
crank  arm,  which  corresponds  to  the  maximum  angular 
velocity  of  the  fly-wheel. 


267.  The  greatest  variation  of  the  angular  velocity  of  the 
fly-wheel. 

Let  aa  be  taken  to  represent  the  least  angular  velocity  of 
the  fly-wheel,  corresponding  to  the  position  CB  of  the  crank 
arm,  and  aa  its  greatest  angular  velocity,  corresponding  to 

the  position  CE ;  then  does  ^-  (a38— a22)  represent  the  work 

accumulated  in  the  fly-wheel  between  these  positions,  which 
accumulated  work  is  equal  to  the  excess  of  that  done  by  the 
power  over  that  expended  upon  the  resistances  whilst  the 
crank  arm  revolves  from  the  one  position  into  the  other, 
and  is  therefore  represented  by  the  difference  of  the  values 
given  to  the  formula  (338)  when  the  two  values  K — ^  and 
i,  determined  by  equation  (340),  are  substituted  in  it  for  0. 
Now  this  difference  is  represented  by  the  formula 

_.       (  m  (TT— 11— *})  ) 

P,#  -j  vers.  (if— ?])— vers.  q  — —    -  r, 

(  /         2>]\ ) 

or  by  Pjfl  |  2  cos.  v—  m  1 1  —  — J  p 

U.T                            (                       /         2'i\  ) 
.  /_.  2        a\ — ~p  Al  a  pno   y, ^,1  i  I   y  . 

••  o    V   s  —    a  /  ~~      i      i        ^v>to.   ') — //ti  j.  if) 

^0'  \  \  it  i     } 

/.a32— a22= — j^  i  2  cos.  n--*»n  —  7)  | (341); 

in  which  equation  *j  is  taken  (equation  340)  to  represent 
that  angle  whose  sine  is  — . 


268.  The  dimensions  of  the  fly-wheel,  such  that  its  angular 
velocity  may  at  no  period  of  a  revolution  deviate  beyond 
prescribed  limits  from  the  mean. 

Let  $N  be  taken  to  represent  the  mean  number  of  revo- 

u 

lutions  made  by  the  fly-wheel  per  minute;  then  will  -J— 

oU 


358  THE   FLY-WHEEL. 

represent  the  mean  number  of  revolutions   or  parts  of  a 

N  NV 

revolution  made  by  it  per  second,  and  fcr^,  or  -^-,  the 

mean  space  described  per  second  by  a  point  in  the  fly-wheel 
whose  distance  from  the  centre  is  unity,  or  the  mean  angular 
velocity  of  the  fly-wheel.  Now,  let  the  dimensions  of  the 
fly  wheel  be  supposed  to  be  such  as  are  sufficient  to  cause 
its  angular  velocity  to  deviate  at  no  period  of  its  revolution 

by  more  than  -th  from  its  mean  value  ;  or  such  that  the  max- 
J  n 

imum  value  as  of  its  angular  velocity  may  equal  -^r  I  1  +  -  I 
and  that  its  minimum  value  a2  may  equal  -^H  1  —  -  I  ;  then 


Substituting  in  equation  (341), 

2*] 


Let  H  be  taken  to  represent  the  horses'  power  of  the 
engine,  estimated  at  its  driving  point  or  piston  ;  then  will 
33000H  represent  the  number  of  units  of  work  done  per 
minute,  upon  the  piston.  But  this  number  of  units  of  work 
is  also  represented  by  %Nm  .  2?^  ;  since  %Nm  is  the  number 
of  strokes  made  by  the  piston  per  minute,  and  2P^  is  the 
work  done  on  the  piston  per  stroke, 

TT 

:.&,a=  6600(%-. 

Era 

Substituting  this  value  for  2P:a  in  the  above  equation,  we 
obtain,  by  reduction, 

66000.30V) 
-  ^      f 

Let  &  be  taken  to  represent  the  radius  of  gyration  of  the 
wheel,  and  M  its  volume;  then  (Art.  80.)  MAf^I,  therefore 
M-M.^2=^I.  But  M-M  represents  the  weight  of  the  wheel 
in  Ibs.  ;  let  W  represent  its  weight  in  tons  ;  therefore, 
aM=2240W.  Substituting  this  value,  and  solving  in 
respect  to  "W, 


THE   FLY-WHEEL.  359 

66000.30^)    <  /,       to\\    H» 


Substituting  their  values  for  *  and  g,  and  determining  the 
numerical  value  of  the  co-efficient, 


W=86491  {  1  cos.  ,-  (l  -  5)  [  g-  .....  (343). 

If  the  influence  of  the  work  accumulated  in  the  arms  of 
the  wheel  be  given  in,  for  an  increase  of  the  equalising 
power  beyond  the  prescribed  limits,  that  accumulated  in  the 
heavy  rim  or  ring  which  forms  its  periphery  being  alone 
taken  into  the  account;*  then  (Art.  86.)  H&9=I=2flrfaK 
(R24~Jc2),  where  b  represents  the  thickness,  c  the  depth,  and 
R  the  mean  radius  of  the  rim.  But  by  Guldiims's  first 
property  (Art.  38.),  2^K=M;  therefore  &2=(R2  +  ic2). 
Substituting  in  equation  (343) 

W=86491  )  1  co,  ,- 

If  the  depth  c  of  the  rim  be  (as  it  usually-  is)  small  as 
compared  with  the  mean  radius  of  the  wheel,  Jcj2  may  be 
neglected  as  compared  with  R2,  the  above  equation  then 
becomes 

i  2  /         2>)\  )    H?& 

W=86491  {  -  cos.  ,-(l  -  -]  |  ^  .  .  .  .  (345)  ; 

by  which  equation  the  weight  "W  in  tons  of  a  fly-wheel  of  a 
given  mean  radius  R  is  determined,  so  that  being  applied  to 
an  engine  of  a  given  horse  power  H,  making  a  given  num- 
ber of  revolutions  per  minute  JN",  it  shall  cause  the  angular 

velocity  of  that  wheel  not  to  vary  by  more  than  -th  from  its 

mean  value.  It  is  to  be  observed  that  the  weight  of  the 
wheel  varies  inversely  as  the  cube  of  the  number  of  strokes 
made  by  the  engine  per  minute,  so  that  an  engine  making 
twice  as  many  strokes  as  another  of  equal  horse  power, 

*  If  the  section  of  each  arm  be  supposed  uniform  and  represented  by  /c,  and 
the  arms  be  p  in  number,  it  is  easily  shown  from  Arts.  79.,  81.,  that  the 
momentum  of  inertia  of  each  arm  about  its  extremity  is  very  nearly  repre- 
sented by  i/c(R—  ic)3,  where  c  represents  the  depth  of  the  rim;  so  that  the 

whole  momentum  of  inertia  of  the  arms  is  represented  by  ^/c(R—  jc1)3,  which 

o 

expression  must  be  added  to  the  momentum  of  the  rim  to  determine  the  whole 
momentum  I  of  the  wheel.  It  appears,  however,  expedient  to  give  the  inertia 
of  the  arms  to  the  equalising  power  of  the  wheeL 


360  THE   FLY-WHEEL. 

would  receive  an  equal  steadiness  of  motion  from  a  fly- 
wheel of  one  eighth  the  weight;  the  mean  radii  of  the 
wheels  being  the  same. 

If,  in  equation  (342),  we  substitute  for  I  its  value  2tffoR3, 
or  2tfKH3  (representing  by  K  the  section  be  of  the  rim),  and 
if  we  suppose  the  wheel  to  be  formed  of  cast  iron  of  mean 
quality,  the  weight  of  each  cubic  foot  of  which  may  be 
assumed  to  be  450  lb.,  we  shall  obtain  by  reduction 


R^=  68521  i-  cos.  ^-(l--)l^...    .(346); 
(  m  \        if  I  j  N  K 

by  which  equation  is  determined  the  mean  radius  R  of  a  fly- 
wheel of  cast  iron  of  a  given  section  K,  which  being  applied 
to  an  engine  of  given  horse  power  H,  making  a  given  num- 
ber of  revolutions  J-N"  per  minute,  shall  cause  its  angular 

velocity  not  to  deviate  more  than  —  th  from  the  mean  ;   or 

conversely,  the  mean  radius  being  given,  the  section  K  may 
be  determined  according  to  these  conditions. 


269.  In  the  above  equations,  m  is  taken  to  represent  the 
number  of  effective  strokes  made  by  the  piston  of  the  engine 
whilst  the  fly-wheel  makes  one  revolution,  and  ?i  to  represent 

<779 

that  angle  whose  sine  is  —  . 

Let  now  the  axis  of  the  fly-wheel  be  supposed  to  be  a 
continuation  of  the  axis  of  the  crank,  so  that  both  turn  with 
the  same  angular  velocity,  as  is  usually  the  case  ;  and  let  its 
application  to  the  single-acting  engine,  the  double-acting 
engine,  and  to  the  double  crank  engine,  be  considered  sepa- 
rately. 

1.  In  the  single-acting  engine,  but  one  effective  stroke  of 
the  piston  is  made  whilst  the  fly-wheel  makes  each  revolution. 

In  this  case,  therefore,  m=l,  and  sin.  *)=—  =0-3183098  = 
sin.  18°  33';  therefore,  cos.  ?]  =  -9480460,  also  -  = 

it 

•103055  ;   therefore,  1  —  —  =  -793888. 


co*  ,--          = 


THE   FLY-WHEEL.  361 

Substituting  in  equations  (345)  and  (346), 

W= 95330-64  ^5, 
rr~ 

•  (347); 

by  which  equations  are  determined,  according  to  the  pro- 
posed conditions,  the  weight  W  in  tons  of  a  fly-wheel  for  a 
single-acting  engine,  its  mean  radius  in  feet  R  being  given, 
and  its  material  being  any  whatever;  and  also  its  mean 
radius  R  in  feet,  its  section  (in  square  feet)  K  being  given, 
and  its  material  being  cast  iron  of  mean  quality ;  and  lastly, 
the  section  K  of  its  rim  in  square  feet,  its  mean  radius  K 
being  given,  and  its  material  being,  as  before,  cast  iron. 

2.  In   the    double-acting  engine,   two   effective    strokes  are 
made    by    the     piston,     whilst    the    fly-wheel    makes     one 

revolution.     In  this  cases  therefore,  m  =  2  and  sin.  ij=-= 

if 

0-636619  =  sin.  39°  32';  therefore,  cos.  *]  =  -7712549  j  = 
39 


O    0' 


/  Oy,    v 

=  -21963  ;  therefore  (  1  -  -^  )  =  '56074  ; 


-  cos.  *i  -    l  -  —       =  -21051. 


m 


-  (l  -  —  )     = 
\          *  i 


Substituting  in  equations  (345)  and  (346), 


24-3593  'EM  T  E.n 

R=—  S—  V  -£,  £=14424^^-  .....  (348); 

by  which  equations  the  weight  of  the  fly-wheel  in  tons,  the 
mean  radius  in  feet,  and  the  section  of  the  rim  in  square 
feet  are  determined  for  the  double-acting  engine. 

3.  In  the  engine  working  with  two  cylinders  and  a  double 
crank,  it  has  been  shown  (Art.  263.)  that  the  conditions  of 
the  working  of  the  two  arms  of  a  double  crank  are  precisely 
the  same  as  though  the  aggregate  pressure  2Pt  upon  their 
extremities,  were  applied  to  the  axis  of  the  crank  by  the 
intervention  of  a  single  arm  and  a  single  connecting  rod; 


362  THE   FKICTION   OF   THE    FLY-WHEEL. 

the  length  of  this  arm  being  represented  by  —  =  instead  of  #, 

1/2 

'  and  its  direction  equally  dividing  the  inclination  of  the  arms 
of  the  double  crank  to  one  another. 

Now,  equations  (345)  and  (346)  show  the  proper  dimen 
sions  of  the  fly-wheel  to  be  wholly  independent  of  the 
length  of  the  crank  arm  ;  whence  it  follows  that  the  dimen- 
sion of  a  fly-wheel  applicable  to  the  double  as  well  as  a 
single  crank,  are  determined  by  those  equations  as  applied 
to  the  case  of  a  double-acting  engine,  the  pressure  upon 
whose  piston  rod  is  represented  by  2Pj.  But  in  assuming 
£Nm  .  ^P1«=S3000H,  we  have  assumed  the  pressure  upon 
the  piston  rod  to  be  represented  by  Pj  ;  to  correct  this  error, 
and  to  adapt  equations  (345)  and  (346)  to  the  case  of  a 
double  crank  engine,  we  must  therefore  substitute  -JH  for  H 
in  those  equations.  "We  shall  thus  obtain 


19-3339 


by  which  equations  the  dimensions  of  a  fly-wheel  necessary 
to  give  the  required  steadiness  of  motion  to  a  double  crank 
engine  are  determined  under  the  proposed  conditions. 


THE  FKICTION  OF  THE  FLY-WHEEL. 

270.  "W  representing  the  weight  of  the  wheel  and  9  the 
limiting  angle  of  resistance  between  the  surface  of  its  axis 
and  that  of  its  bearings,  sin.  <p  will  represent  its  coefficient 
of  friction  (Art.  138.),  and  "W"  sin.  <p,  the  resistance  opposed 
to  its  revolution  by  friction  at  the  surface  of  its  axis.  Now, 
whilst  the  wheel  makes  one  revolution,  this  resistance  is 
overcome  through  a  space  equal  to  the  circumference  of  the 
axis,  and  represented  by  2tfp,  if  p  be  taken  to  represent  the 
radius  of  the  axis.  The  work  expended  upon  the  friction  of 
the  axis,  during  each  complete  revolution  of  the  wheel,  is 
therefore  represented  by  2^pW  sin.  <p ;  and  if  IS"  represent 
the  number  of  strokes  made  by  the  engine  per  minute,  and 

vr 

therefore  — the  number  of  revolutions  made  by  the  fly-wheel 
2 


MODULUS    OF   THE    CRANK   AND   FLY-WHEEL.  363 

per  minute,  then  will  the  number  of  units  of  work  expended 
per  minute,  upon  the  friction  of  the  axis  be  represented  by 
N*pW  sin.  9 ;  and  the  number  of  horses'  power,  or  the  frac- 
tional part  of  a  horse's  power  thus  expended,  by 

.  (350). 


33000 

If  in  this  equation  we  substitute  for  W  the  weight  in  Ibs. 
of  the  fly-wheel  necessary  to  establish  a  given  degree  of 
steadiness  in  the  engine,  as  determined  by  equations  (347), 
(348),  and  (349),  we  shall  obtain  for  the  horse  power  lost  by 
friction  of  the  fly-wheel,  in  the  single-acting  engine,  the 
double-acting  engine,  and  the  double  crank  engine,  respec- 
tively, the  formulae 


THE  MODULUS  OF  THE  CRANK  AND  FLY-WHEEL. 

271.  If  Sj  represent  the  space  traversed  by  the  piston  of 
the  engine  in  any  given  time,  and  a  the  radius  of  the  crank, 
W  the  weight  of  the  fly-wheel  in  Ibs.,  and  p  the  radius  of  its 

Q 

axis,  then  will  2a  represent  the  length  of  each  stroke,  —  the 

2$ 

Ql 

number  of  strokes  made  in  that  time,  and  %fpW  sin.  <p  .  —  * 

' 


or  if  WSj      sin.  <p  the  work  expended  upon  the  friction  of  the 
a 

fly-wheel  during  that  time,  which  expression  being  added  to 
the  equation  (329)  representing  the  work  necessary  to  cause 
the  crank  to  yield  a  given  amount  of  work  U2  to  the  ma- 
chine driven  by  it  (independently  of  the  work  expended  on 
the  friction  of  the  fly-wheel),  will  give  the  whole  amount  of 
work  which  must  be  done  upon  the  combination  of  the  crank 
and  fly-wheel,  to  cause  this  given  amount  of  work  to  be 
yielded  by  it,  on  the  machine  which  tiie  crank  drives.  Let 
this  amount  of  work  be  represented  by  U^  then  in  the  case 
in  which  the  directions  of  the  driving  pressure  and  the  re- 
sistance upon  the  crank  are  parallel  (equation  (329),  and  the 


364 


THE  GOVERNOR. 


friction  of  the  crane  guide  is  neglected,  we  obtain  for  the 
modulus  of  the  crank  and  fly-wheel  in  the  double-acting 
engine 

U  =  |  1  +  £fc  sin.  9,  + 1«  sin.  9,  )  }  U,  +  *WS,  f  sin.  9  (352). 
(          2\#  a  /  )  a 


THE  GOVERNOR. 

272.  This  instrument  is  represented  in  the  figure,  under 
that  form  in  which  it  is  most  commonly  applied  tp  the  steam 
engine.  BD  and  CE  are  rods  jointed 
at  A  upon  the  vertical  spindle  AF, 
and  at  D  and  E  upon  the  rods  DP 
and  EP,  which  last  are  again  jointed 
at  their  extremities  to  a  collar  fitted 
accurately  to  the  surface  of  the  spin- 
dle and  moveable  upon  it.  At  their 
extremities  B  and  C,  the  rods  DB 
and  EC  carry  two  heavy  balls,  and 
being  swept  round  by  the  spindle  — 
which  receives  a  rapid  rotation  al- 
ways proportional  to  the  speed  of  the 
machine,  whose  motion  the  governor 
is  intended  to  regulate — these  arms 
by  their  own  centrifugal  force,  and 
that  of  the  balls,  are  made  to  separate,  and  thereby  to  cause 
the  collar  at  P  to  descend  upon  the  spindle,  carrying  with  it, 
by  the  intervention  of  the  shoulder,  the  extremity  of  a  lever, 
whose  motion  controls  the  access  of  the  moving  power  to 
the  driving  point  of  the  machine,  closing  the  throttle  valve 
and  shutting  off  the  steam  from  the  steam  engine,  or  closing 
the  sluice  and  thus  diminishing  the  supply  of  water  to  the 
water-wheel.  Let  P  be  taken  to  represent  the  pressure  of 
the  extremity  of  the  lever  upon  the  collar,  Q  the  strain 
thereby  produced  upon  each  of  the  rods  DP  and  EP  in  the 
direction  of  its  length,  W  the  weight  of  each  of  the  balls,  w 
the  weight  of  each  of  the  rods  BD  and  CE,  AB=a,  AD=&, 
DP=c,  FAB=d,  APDr=^.  Now  upon  either  of  these  rods 
as  BD,  the  following  pressures  are  applied :  the  weight  of 
the  ball  and  the  weight  of  the  rod  acting  vertically,  the 
centrifugal  force  of  the  ball  and  the  centrifugal  force  of  the 
rod  acting  horizontally,  the  strain  Q  of  the  rod  DP,  and 
the  resistance  of  the  axis  A.  If  a  represent  the  angular 


THE  GOVERNOR.  365 

W  "W 

velocity  of  the  spindle,  —  aa  .  FB,  or  —  a90sinJ,  willrepre- 

y  & 

sent  the  centrifugal  force  upon   the   ball   (equation  102), 

and  —  »V  sin.  6  cos.  &  its  moment  about  the  point  A  ;  also 

9 

the  centrifugal  force  of  the  rod  BD  produces  the  same  effect 
as  though  its  weight  were  collected  in  its  centre  of  gravity 
(Art.  124.),  whose  distance  from  A  is  represented  by  \(ci—b\ 
so  that  the  centrifugal  force  of  the  rod  is  represented  by 

ID 

J—  ot?(a—  5)  sin.  6,  and  its  moment  about  the  point  A  by 

1JO 

J—  a*  (a—  5)3  sin.  &  cos.  6.  On  the  whole,  therefore,  the  sum  of 
the  moments  of  the  centrifugal  forces  of  the  rod  and  ball  are 
represented  by  —  {~Wa?-\-%w(a—  5)3j  sin.  6  cos.  6.  Now  if  f* 

t/ 

represent  the  weight  of  each  unit  in  the  length  of  the  rod, 
w  =  f*(»+  &)  ;  therefore  Wa*  +  %w(a  -  &)3  =  W  a9 
(a—  b).     Let  this  quantity  be  represented  by  " 


-       («-5)  ....  (353); 


then  will  —  "W^*  sin.  6  cos.  6  represent  the  sum  of  the  moments 

9 

of  the  centrifugal  forces  of  the  rod  and  ball  about  A.  More- 
over, the  sum  of  the  moments  of  the  weights  of  the  rod  and 
ball,  about  the  same  point,  is  evidently  represented  by  Wa 
sin.  b  +  w$(a—  V)  sin.  6,  or  by  \Wa+^(a?—  52)}  sin.  6  ;  let  this 
quantity  be  represented  by  W2#  sin.  6, 


Also  the  moment  of  Q  about  A=Q  .  AH=Q3  sin. 
Therefore,  by  the  principle  of  the  equality  of  moments,  ob- 
serving that  the  centrifugal  force  of  the  rod  and  ball  tend  to 
communicate  motion  in  an  opposite  direction  from  their 
weights  and  the  pressure  Q, 


—  WX  sin.  6  cos.  t=Qb  sin.  (d+fiJ+Wji  sin.  6. 


360 


THE  GOVERNOR. 


Now  P  is  the  resultant  of  the  pressures  Q  acting  in  the 
directions  of  the  rods  PD  and  PE,  and  inclined  to  one 
another  at  the  angle  20,  ;  therefore  (equation  13), 

P=2Q  cos.  6l  ; 


.'.  Q  sin.  (/)  +  &,)  =  JP       --=frp  jsin.  6  +  cos.  6  tan.  ^  . 

COS.  fig 

But  since  the  sides  5  and  <?  of  the  triangle  APD  are  oppo- 

site to  the  angles  6    and  6,  therefore   sin*   1=-  ;   therefore 

sin.  A      c 

cos  ' 


.  6t=  (l-l'sin.2*)  V; 


=          sn.   +  -  sn.    cos.          —    - 


Substituting  this  value  in  the  preceding  equation,  dividing 
by  sin.  0,  and  writing  (1—  cos.  2d)  for  sin.  2$,  we  obtain 


aa.  ..(355); 


which  equation,  of  four  dimensions  in  terms  of  cos.  0,  being 
solved  in  respect  to  that  variable,  determines  the  inclination 
of  the  arms  under  a  given  angular  velocity  of  the  spindle. 
It  is,  however,  more  commonly  the  case  that  the  inclination 
of  the  arms  is  given,  and  that  the  lengths  of  the  arms, 
or  the  weights  of  the  balls,  are  required  to  be  determined, 
so  that  this  inclination  may,  under  the  proposed  conditions, 
be  attained.  In  this  case  the  values  of  W,  and  W2  must  be 
substituted  in  the  above  equation  from  equations  (353)  and 
(354),  and  that  equation  solved  in  respect  to  a  or  "W. 

The  values  of  b  and  c  are  determined  by  the  position  on 
the  spindle,  to  which  it  is  proposed  to  make  the  collar 
descend,  at  the  given  inclination  of  the  arms  or  value  of  0. 
If  the  distance  AP,  of  this  position  of  the  collar  from  A,  be 
represented  by  ^,  we  have  h=b  cos.  b-\-e  cos.  ^, 


-^$m.*4  V  ----  (356); 


THE   GOVERNOR. 


367 


from  which  equation  and  the  preceding,  the  value  of  one 
of  the  quantities  1)  or  c  may  be  determined,  according  to  the 
proposed  conditions,  the  value  of  the  other  being  assumed  to 
be  any  whatever. 

If  N"  represent  the  number  of  revolutions,  or  parts  of  a 
revolution,  made  per  second  by  the  fly-wheel,  and  /N  the 
number  of  revolutions  made  in  the  same  time  by  the  spindle 
of  the  governor,  then  will  fyirylS  represent  the  space  a  de- 
scribed per  second  by  a  point,  situated  at  distance  unity  from 
the  axis  of  the  spindle.  Substituting  this  value  for  a  in 
equation  (355),  and  assuming  &—  <?,  we  obtain 

4ffV2N8 

-?—  WXcos.  0=PJ+W2a  ____  (357)  : 
t/ 

also  by  equation  (356), 

h=2b  cos.  A  .....  (358). 

Eliminating  cos.  d  between  these  equations,  and  solving  in 
respect  to  A, 


Let  P  (1-f™)  and  P  (l-~)  represent  the  values  of  P 
corresponding  to  the  two  states  bordering  upon  motion 
(Art.  140)  and  let  N  (l  +  £)  and  N  (1—^)  be  the  correspond- 
ing values  of  N  ;  so  that  the  variation  either  way  of  ^th  from 
the  mean  number  N"  of  revolutions,  may  be  upon  the  point 
of  causing  the  valve  to  move.  If  these  values  be  respectively 
substituted  for  P  and  N  in  the  above  formula,  it  is  evident 
that  the  corresponding  values  of  fi  will  be  equal.  Equating 
those  values  of  h  and  reducing,  we  obtain 


By  which  equation  there  is  established  that  relation  between 
the  quantities  W2,  &,  P,  m  which  must  obtain,  in  order  that  a 
variation  of  the  number  of  revolutions,  ever  so  little  greater 


368 


THE   CAKKIAGE-WHEEL. 


than  the  ^th  part,  may  cause  the  valve  to  move.    Neglect- 
ing \  as  small  when  compared  with  n. 


/       w 

I  Jl  •  — 

which  expression,  representing  that  fractional  variation  in  the 
number  of  revolutions  which  is  sufficient  to  give  motion  to 
the  valve,  is  the  true  measure  of  the  SENSIBILITY  of  the 
governor. 


273.  The  joints  E  and  D  are  sometimes 
fixed  upon  the  arms  AB  and  AC  as  in  the 
accompanying  figure,  instead  of  upon  the 
prolongations  of  those  arms  as  in  the  pre- 
ceding figure.  All  the  formulae  of  the 
last  Article  evidently  adapt  themselves 
to  this  case,  if  b  be  assumed  =0  (in  equa- 
tions 353,  354).  The  centrifugal  force  of 
the  rods  EP  and  DP  is  neglected  in  this 
computation. 


THE  CARRIAGE-WHEEL. 

274:.  Whatever  be  the  nature  of  the  resistance  opposed  to 
the  motion  of  a  carriage- wheel,  it  is  evidently  equivalent  to 
that  of  an  obstacle,  real  or  imaginary,  which  the  wheel  may 
be  supposed,  at  every  instant,  to  be  in  the  act  of  surmount- 
ing. Indeed  it  is  certain,  that,  however  yielding  may  be  the 
material  of  the  road,  yet  by  reason  of  its  compression  before 
the  wheel,  such  an  immoveable  obstacle,  of  exceedingly  small 
height,  is  continually  in  the  act  of  being  presented  to  it. 


275.  The  two-wheeled  carriage. 

Let  AB  represent  one  of  the  wheels  of  a  two-wheeled 
carriage,  EF  an  inclined  plane,  which  it  is  in  the  act  of  as- 
cending, O  a  solid  elevation  of  the  surface  of  the  plane,  or  an 
obstacle  which  it  is  at  any  instant  in  the  act  of  passing  over, 


THE   CAKRIAGE-WHEEL. 


369 


P  the  corresponding  trac- 
tion, W  the  weight  of  the 
wheel  and  of  the  load  which 
it  supports. 

Now  the  surface  of  the  lox 
of  the  wheel  being  in  the 
state  bordering  upon  motion 
on  the  surface  of  the  axle, 
the  direction  of  the  resist- 
ance of  the  one  upon  the 
other  is  inclined  at  the  limit- 
ing angle  of  resistance,  to  a 
radius  of  the  axle  at  their, 
point  of  contact  (Art.  14rl.)i. 
This  resistance  has,,  more- 
over, its  direction;  through 
the  point  of  contact  O  of 
the  tire  of  the  wheel  with  the  obstacle  on  which  it  is  in  the 
act  of  turning.  If,  therefore,  OK  be  drawn  intersecting  the 
circumference  of  the  axis  in  a  point  <?,  such  that  the  angle 
CcR  may  equal  the  limiting  angle  of  resistance  <p,,  then  will 
its  direction  be  that  of  the  resistance  of  the  obstacle  upon 
the  wheel. 

Draw  the  vertical  GH  representing  the  weight  "W,  and 
through  H  draw  HK  parallel  to  OK,  then  will  this  line 
represent  (to  the  same  scale)  the  resistance  K,  and  GK  the 
traction  P  (Art.  14.)  ; 

,    P  _GK    sin.  GHK  sin..  GHK 

'  W~GH-sm.  GKH-sin.  (PGH-GHK)= 

sin.  WsO 
sin.  (PLW-W*Oy 

Let  K= radius  of  wheel,  p= radius  of  axle,  AGO =77,  ACTW 
— i=inclination  of  the  road  to  the  horizon,  ^inclination  of 
direction  of  the  traction  to  the  road.  Now  WsO=WCO-f- 


COs,  but 


=*  +  iy,  and 


Let  CO*  be  re- 


presented by  a,  then  WsO=i 

sin.  a^^sin.  9  ....  (360). 


Also PLW=^+*+d;  therefore ~PLW-WsO=:--(ri+a-6)-9 

24: 


3TO  THE   CARRIAGE-WHEEL. 


.-.P=W-  ^r-^ (361); 

COS.  (*)+ a  — 6) 

when  the  direction  of  traction  is  parallel  to  the  road,  d=0, 
.•.P=W{sin.  1  +  cos.  i  tan.  (*i+a)}  ....  (362). 

If  the  road  and  the  direction  of  traction  be  both  horizontal 
6=i=Q,  and 

P=W  tan.  (ij+a) (363). 

In  all  cases  of  traction  with  wheels  of  the  common  dimen- 
sions upon  ordinary  roads,  AGO  or  v\  is  an  exceedingly  small 
angle ;  a  is  also,  in  all  cases,  an  exceedingly  small  angle 
(equation  360);  therefore  tan.  (77+  a)— 77  +  a  very  nearly. 
Now  if  A  be  taken  to  represent  the  arc  AO,  whose  length 
is  determined  by  the  height  of  the  obstacle  and  the  radius 
of  the  wheel,  then 

1=5 (364)- 

Substituting  the  value  of  a  from  equation  (360), 

p    w  (A  +  p  sin.  9)  , 

P=W.±-    -^—     i (365). 

276.  It  remains  to  determine  the  value  of  the  arc  A  inter- 
cepted between  the  lowest  point  to  which  the  wheel  sinks  in 
the  road,  and  the  summit  0  of  the  obstacle,  which  it  is  at 
every  instant  surmounting.  Now,  the  experiments  of  Cou- 
lomb, and  the  more  recent  experiments  of  M.  Morin,*  ap- 
pear to  have  fully  established  the  fact,  that,  on  horizontal 
roads  of  uniform  quality  and  material,  the  traction  P,  when 
its  direction  is  horizontal,  varies  directly  as  the  load  W,  and 
inversely  as  the  radius  B,  of  the  wheel ;  whence  it  follows 
(equation  365),  that  the  arc  A  is  constant,  or  that  it  is  the 
same  for  the  same  quality  of  road,  whatever  may  be  the 
weight  of  the  load,  or  the  dimensions  of  the  wheel.f  The 

*  Experiences  sur  le  Tirage  des  Voitures,  faites  en  1837  et  1888.  (See  AP- 
PENDIX.) 

f  In  explanation  of  this  fact  let  it  be  observed,  that  although  the  wheel 
sinks  deeper  beneath  the  surface  of  the  road  as  the  material  is  softer,  yet  the 
obstacle  yields,  for  the  same  reason,  more  under  the  pressure  of  the  wheel,  the 
arc  A  being  by  the  one  cause  increased,  and  by  the  other  diminished.  Also, 
that  although  by  increasing  the  diameter  of  the  wheel  the  arc  A  would  be  ren- 
dered greater  if  the  wheel  sank  to  the  same  depth  as  before,  yet  that  it  does 
not  sink  to  the  same  depth  by  reason  of  the  corresponding  increase  of  the  sur- 
face which  sustains  the  pressure. 


THE   CAKKI  AGE-WHEEL.  371 

constant  A  may  therefore  be  taken  as  a  measure  of  the  re- 
sisting quality  of  the  road,  and  may  be  called  the  modulus 
of  its  resistance. 

The  mean  value  of  this  modulus  being  determined  in  re- 
spect to  a  road,  whose  surface  is  of  any  given  quality,  the 
value  of  7)  will  be  known  from  equation  (364),  and  the  rela- 
tion between  the  traction  and  the  load  upon  that  road,  under 
all  circumstances  ;  it  being  observed,  that,  since  the  arc  A 
is  the  same  on  a  horizontal  road,  whatever  be  the  load,  if  the 
traction  be  parallel,  it  is  also  the  same  under  the  same  cir- 
cumstances upon  a  sloping  road  ;  the  effect  of  the  slope  be- 
ing equivalent  to  a  variation  of  the  load.  The  same  substi- 
tution may  therefore  be  made  for  tan.  (?)+a)  in  equation 
(362),  as  was  made  in  equation  (363), 


=         sn.  <+  cos. 


277.  The  lest  direction  of  traction  in  the  two-wheeled 
carriage. 

This  best  direction  of  traction  is  evidently  that  which  gives 
to  the  denominator  of  equation  (361)  its  greatest  value  ;  it 
is  therefore  determined  by  the  equation 

....  (367). 


278.  The  four-wheeled  carriage. 

Let  W1?  Wa  represent  the  loads  borne  by  the  fore  and 
hind  wheels,  together  with  their  own  weights,  K0  Ka  their 
radii,  p,,  p2  the  radii  of  their  axles,  and  <pl5  <pa  the  limiting  an- 
gles of  resistance.  Suppose  the  direction  of  the  traction  P 
parallel  to  the  road,  then,  since  this  traction  equals  the  sums 
of  the  tractions  upon  the  fore  and  hind  wheels  respectively, 
we  have  by  equation  (366) 


372  THE   CAKRIAGE-WHEEL. 

or 

3  sn. 


^)  sin.  9l+W2(^)sin.92  jcos.  •  .  .  .  (368). 


279.  The  work  accumulated  in  the  carriage-wheel.* 

Let  I  represent  the  moment  of  inertia  of  the  wheel  about 
its  axis  and  M  its  volume;  then  will  MR2+I  represent  its 
moment  of  inertia  (Art.  79.)  about  the  point  in  its  circum- 
ferences about  which  it  is,  at  every  instant  of  its  motion,  in 
the  act  of  turning.  If,  therefore,  a  represent  its  angular 
velocity  about  this  point  at  any  instant,  U  the  work  at  that 
instant  accumulated  in  it,  and  p  the  weight  of  each  cubical 

unit  of  its  mass,  then  (Art.  75.),  U=Jaa-(MRa+I)  =  |-M 

y  y 

M* 

(aR)a-f-Jaa-I.     Now  if  Y  represent  the  velocity  of  the  axis 
of  the  wheel,  aE=Y; 


whence  it  follows,  that  the  whole  work  accumulated  in  the 
rolling  wheel  is  equal  to  the  sum  obtained  by  adding  the 
work  which  would  have  been  accumulated  in  it  if  it  had 
moved  with  its  motion  of  translation  only,  to  that  which 
would  have  been  accumulated  in  it  if  it  had  moved  with  its 
motion  of  rotation  only.  If  we  represent  the  radius  of  gyra- 
tion (Art.  80.)  by  }£,  I=MKa  ;  whence  substituting  and 
reducing, 

°       .....  (369). 


The  accumulated  work  is  therefore  the  same  as  though  the 
wheel  had  moved  with  a  motion  of  translation  only,  but  with 

a  greater  velocity,  represented  by  the  expression  1  1  +  -pa  1  Y. 

*  For  a  further  discussion  of  the  conditions  of  the  rolling  of  a  wheel,  see  a 
paper  in  the  Appendix  on  the  Rolling  Motion  of  a  Cylinder. 

f  The  angular  velocity  of  the  wheel  would  evidently  be  a,  if  its  centre  were 
fixed,  and  its  circumference  made  to  revolve  with  the  same  velocity  as  now. 


ACCELERATED   OE  RETARDED  MOTION.  373 


280.    ON   THE   STATE   OF   THE   ACCELERATED   OR   THE   RETARDED 
MOTION   OF  A  MACHINE. 

Let  the  work  Ul  done  upon  the  driving  point  of  a  machine 
be  conceived  to  be  in  excess  of  that  U3  yielded  upon  the 
working  points  of  the  machine  and  that  expended  upon  its 
prejudicial  resistances.  Then  we  have  by  equation  (117) 

11,=  AU.+  BS,  +^(V°-  V^tox'  ; 

where  Y  represents  the  velocity  of  the  driving  point  of  the 
machine  after  the  work  'Ul  has  been  done  upon  it,  Y4  that 
when  it  began  to  be  done,  and  2i0Xa  the  coefficient  of  equable 
motion.  ]Now  let  Sx  represent  the  space  through  which  Uj 
is  done,  and  S2  that  through  which  U  ,  is  done  ;  and  let  the 
above  equation  be  differentiated  in  respect  to  Sn 

.«ZU,_    dU.    dB  1    dV 

••dS,-    dS,  '  3§l+"*9r<m**  *  , 

but   TT"1  —  PI  (Art.  51.)  if  P,  represent  the  driving  pressure. 


Also  -SOT  =  Pa,  if  Pa  represent  the  working  pressure  ;  also 

'    dV     dt  dV     1       dV 

V''-----=f  (equation   72). 


If,  therefore,  we  represent  by  A  the  relation  -1,  between  the 


spaces  described  in  the  same  exceedingly  small  time  by  the 
driving  and  working  points,  we  have 

-2wXa  .....  (370); 

9 


where/"  (Art.  95.)  represents  the  additional  velocity  actually 
acquired  per  second  by  the  driving  point  of  the  machine,  if 
Pj  and  P2  be  constant  quantities,  or,  if  not,  the  additional 
velocity  which  would  be  acquired  in  any  given  second,  if 
these  pressures  retained,  throughout  that  second,  the  values 
which  they  had  at  its  commencement. 


374         THE  ACCELERATION  OR  RETARDATION 


281.  To  determine  the  coefficient  of  equable  motion. 


a  represents  the  sum  of  the  weights  of  all  the  moving 
elements  of  the  machine,  each  being  multiplied  by  the  ratio 
of  its  velocity  to  that  of  the  driving  point,  which  sum  has 
been  called  (Art.  151.)  the  coefficient  of  equable  motion.  If 
the  motion  of  each  element  of  the  machine  takes  place  #bout 
a  fixed  axis,  and  a1}  a,n  «„  &c.,  represent  the  perpendiculars 
from  their  several  axes  upon  the  directions  in  which  they 
receive  the  driving  pressures  of  the  elements  which  precede 
them  in  the  series,  and  ^,  J2,  Z>3,  &c.,  the  similar  perpen- 
diculars upon  the  tangents  to  their  common  surfaces  at  the 
points  where  they  drive  those  that  follow  them  ;  then, 
while  the  first  driving  point  describes  the  small  space  AS15 
the  point  of  contact  of  the  p\h  and  p  +  1th  elements  of  the 
series  will  be  made  (Art.  234.)  to  describe  a  space  repre- 
sented b 


**M    ^ 

a^a^ ...  ct>p 

so  that  the  angular  velocity  of  the  ^?th  element  will  be 
represented  by 


. 

ap 


and  the  space  described  by  a  particle  situated  at  distance  f 
from  the  axis  of  that  element  by 


and  the  ratio  X  of  this  space  to  that  described  by  the  driving 
point  of  the  machine  will  be  represented  by 


a.t 


The  sum  2wX2  wiH  therefore  be  represented  in  respect  to 
this  one  element  by 


Or  if  Ip  represent  the  moment  of  inertia  of  the  element,  and 
PP  the  weight  of  each  cubic  unit  of  its  mass,  that  portion  of 
the  value  of  2wXa  which  depends  upon  this  element  will  be 
represented  by 


OF   THE   MOTION   OF   A   MACHINE.  375 


And  the  same  being  true  of  every  other  element  of  the 
machine,  we  have 


which  is  a  general  expression  for  the  coefficient  of  equable 
motion  in  the  case  supposed.  The  value  of  A  in  equation 
(3T1)  is  evidently  represented  by 


To  determine  the  pressure  upon  the  point  of  contact  of 
any  two  elements  of  a  machine  moving  with  an  accelerated 
or  retarded  motion. 

Let  j?4  be  taken  to  represent  the  resistance  upon  the  point 
of  contact  of  the  first  element  with  the  second,  j92  that  upon 
the  point  of  contact  of  the  second  element  of  the  machine 
with  the  third,  and  so  on.  Then  by  equation  (3TO),  observ- 
ing that,  P!  and  pl  representing  pressures  applied  to  the 
same  element,  ^w^  is  to  be  taken  in  this  case  only  in 
respect  to  that  element,  so  that  it  is  represented  by  p.,!^ 

whilst  A  is  in  this  case  represented  by  —  ,  we  have,  neglect- 
ing friction, 


Substituting  the  value  of  f  from  equation  (371),  and  solving 
in  respect  to  j?,, 


_«!  p    _«!  /   p     _  Ap    \  j^L 

—  b,1  "!>,[    l          V  2wK 


where  the  value  of  A  is  determined  by  equation  (373),  and 
that  of  2i0Xa  by  equation  (37^).  Proceeding  similarly  in 
respect  to  the  second  element,  and  observing  that  the 
impressed  pressures  upon  that  element  are  jpl  and  p»  we 
have 


S76  ACCELERATED   OR   RETARDED   MOTION. 

fi  representing  the  additional  velocity  per  second  of  the 
point  of  application  of  p»   which  evidently  equals  —  /. 

Substituting,  therefore,  the  value  of/  from  equation  (3T1) 
as  before, 


Substituting  the  value  of  ^  from  equation  (374),  and  solv- 
ing in  respect  tojpa,  we  have 


And  proceeding  similarly  in  respect  to  the  other  points  of 
contact,  the  pressure  upon  each  may  be  determined.  It  is 
evident,  that  by  assuming  values  of  A  and  B  in  equations 
(370)  and  (371)  to  represent  the  coefficients  of  the  moduli  in 
respect  to  the  several  elements  of  the  machine,  and  to  the 
whole  machine,  the  influence  of  friction  might,  by  similar 
steps,  have  been  included  in  the  result. 


PART  IV. 


THEOEY  OF  THE  STABILITY  OF  STKUCTUKES. 


GENERAL  CONDITIONS  OF  THE  STABILITY  OF  A  STRUCTURE  OF 
UNCEMETED  STONES.* 

A  STRUCTURE  may  yield,  under  the  pressures  to  which  it  is 
subjected,  either  by  the  slipping  of  certain  of  its  surfaces  of 
contact  upon  one  another,  or  by  their  turning  over  upon  the 
edges  of  one  another ;  and  these  two  conditions  involve  the 
whole  question  of  its  stability. 


THE  LINE  OF  KESISTANCE. 

283.  Let  a  structure  MNLK,  composed  of  a  single  row  of 
uncemented  stones  of  any  forms, 
s  and  placed  under  any  given  circum- 
f  stances  of  pressure,  be  conceived  to 
£,  be  intersected  by  any  geometrical 
surface  1  2,  and  let  the  resultant  a  A 
of  all  the  pressures  which  act  upon 
one  of  the  parts  MN21,  into  which 
this  intersecting  surface  divides  the 
structure,  be  imagined  to  be  taken. 
Conceive,  then,  this  intersecting 
surface"  to  change  its  form  and  posi- 
tion so  as  to  coincide  in  succession 
with  all  the  common  surfaces  of 
contact  8  4,  5  6,  T  8,  9  10,  of  the 
stones  which  compose  the  structure  : 
and  let  £R,  cO,  dD,  eE  be  the  re- 

*  Extracted  from  a  memoir  on  the  Theory  of  the  Arch  by  the  author  of  this 
work  in  the  first  volume  of  the  "  Theoretical  and  Practical  Treatise  on  Bridges," 
by  Professor  Hosking  and  Mr.  Hann  of  King's  College,  published  by  Mr.  Weale. 
These  general  conditions  of  the  equilibrium  of  a  system  of  bodies  in  contact 
were  first  discussed  by  the  author  in  the  fifth  and  sixth  volumes  of  the  "  Cam- 
bridge  Philosophical  Transactions." 

871 


378  THE   LINE   OF   RESISTANCE. 

sultan ts,  similarly  taken  with  &A,  which  correspond  to  these 
several  planes  of  intersection. 

In  each  such  position  of  the  intersecting  surface,  the  result- 
ant spoken  of  having  its  direction  produced,  will  intersect 
that  surface  either  within  the  mass  of  the  structure,  or,  when 
that  surface  is  imagined  to  be  produced,  without  it.  If  it 
intersect  it  without  the  mass  of  the  structure,  then  the  whole 
pressure  upon  one  of  the  parts,  acting  in  the  direction  of 
this  resultant,  will  cause  that  part  to  turn  over  upon  the 
edge  of  its  common  surface  of  contact  with  the  other  part ; 
if  it  intersect  it  within  the  mass  of  the  structure,  it  will  not. 

Thus,  for  instance,  if  the  direction  of  the  resultant  of  the 
forces  acting  upon  the  part  NM  1  2  had  been  &'A',  not  inter- 
secting the  surface  of  contact  1  2  within  the  mass  of  the 
structure,  but  when  imagined  to  be  produced  beyond  it  to  a' ; 
then  the  whole  pressure  upon  this  part  acting  in  a' A!  would 
have  caused  it  to  turn  upon  the  edge  2  of  the  surface  of  con- 
tact 1  2 ;  and  similarly  if  the  resultant  had  been  in  a"  A", 
then  it  would  have  caused  the  mass  to  revolve  upon  the 
edge  1.  The  resultant  having  the  direction  #A,  the  mass 
will  not  be  made  to  revolve  on  either  edge  of  the  surface  of 
contact  1  2. 

Thus  the  condition  that  no  two  parts  of  the  mass  should  be 
made,  by  the  insistent  pressures,  to  turn  over  upon  the  edge 
of  their  common  surface  of  contact,  is  involved  in  this  other, 
that  the  direction  of  the  resultant,  taken  in  respect  to  every 
position  of  the  intersecting  surface,  shall  intersect  that  sur- 
face actually  within  the  mass  of  the  structure. 

If  the  intersecting  surface  be  imagined  to  take  up  an  infi- 
nite number  of  different  positions,  1  2,  3  4,  5  6,  &c.,  and  the 
intersections  with  it,  #,  J,  <?,  d,  &c.,  of  the  directions  of  all 
the  corresponding  resultants  be  found,  then  the  curved  line 
cikcdef,  joining  these  points  of  intersection,  may  with  pro- 
priety be  called  the  LINE  OF  RESISTANCE,  the  resisting  points 
of  the  resultant  pressures  upon  the  contiguous  surfaces  lying 
all  in  that  line.  f>. 

This  line  can  be  completely  determined  by  the  methods  of 
analysis,  in  respect  to  a  structure  of  any  given  geometrical 
form,  having  its  parts  in  contact  by  surfaces  also  of  given 
geometrical  forms.  And,  conversely,  the  form  of  this  line 
being  assumed,  and  the  direction  which  it  shall  have  through 
any  proposed  structure,  the  geometrical  form  of  that  struc- 
ture may  be  determined,  subject  to  these  conditions ;  or 
lastly,  certain  conditions  being  assumed,  both  as  it  regards 
the  form  of  the  structure  and  its  line  of  resistance,  all  that  is 


THE   LINE   OF    PRESSURE. 


379 


necessary  to  the  existence  of  these  assumed  conditions  may 
be  found.  Let  the  structure  ABCD  have  for  its  line  of  re- 
sistance the  line  PQ.  Now 
it  is  clear  that  if  this  line 
cut  the  surface  MN  of  any 
section  of  the  mass  in  a  point 
n  without  the  surface  of  the 
mass,  then  the  resultant  of 
the  pressures  upon  the  mass 
CMN  will  act  through  n, 
and  cause  this  portion  of  the 
mass  to  revolve  about  the 
nearest  point  IS"  of  the  in- 
tersection of  the  surface  of 
secton  K  with  the  surface  of  the  structure. 

Thus,  then,  it  is  a  condition  of  the  equilibrium  that  the 
line  of  resistance  shall  intersect  the  common  surface  of  con- 
tact of  each  two  contiguous  portions  of  the  structure  actually 
within  the  mass  of  the  structure  /  or,  in  other  words,  that  it 
shall  actually  go  through  each  joint  of  the  structure,  avoid- 
ing none :  this  condition  being  necessary,  that  no  two  po5> 
tions  of  the  structure  may  revolve  on  the  edges  of  their 
common  surface  of  contact. 


THE  LINE  OF  PRESSURE. 

» 

284.  But  besides  the  condition  that  no  two  parts  of  the 
structure  should  turn  upon  the  edges  of  their  common  sur- 
faces of  contact,  which  condition  is  involved  in  the  determi- 
nation of  the  LINE  OF  RESISTANCE,  there  is  a  second  condition 
necessary  to  the  stability  of  the  structure.  Its  surfaces  of 
contact  must  no  where  slip  upon  one  another.  That  this 
condition  may  obtain,  the  resultant  corresponding  to  each 
surface  of  contact  must  have  its  direction  within  certain 
limits.  These  limits  are  denned  by  the  surface  of  a  right 
cone  (Art.  1.39.),  having  the  normal  to  the  common  surface 
of  contact  vat  the  above-mentioned  point  of  intersection  of 
the  resultant)  for  its  axis,  and  having  for  its  vertical  angle 
twice  that  whose  tangent  is  the  co-emcient  of  friction  of  the 
surfaces.  If  the  direction  of  the  resultant  be  within  this 
cone,  the  surfaces  of  contact  will  not  slip  upon  one  another ; 
if  it  be  without  it,  they  will. 

Thus,  then,  the  directions  of  the  consecutive  resultants  in 


380  *  THE   STABILITY   OF   A   SOLID   BODY. 

respect  to  the  normal  to  the  point,  where  each  intersects  its 
corresponding  surface  of  contact,  are  to  be  considered  as  im- 
portant elements  of  the  theory. 

Let  then  a  line  ABODE  be  taken,  which  is  the  locus  of 
the  consecutive  intersections  of  the 
resultants  # A,  &B,  cC,  dD,  &c.  The 
direction  of  the  resultant  pressure 
upon  every  section  is  a  tangent  to 
this  line ;  it  may  therefore  with  pro- 
priety be  called  the  LINE  OF  PRESSURE. 
Its  geometrical  form  may  be  deter- 
mined under  the  same  circumstances 
as  that  of  the  line  of  resistance.  A 
straight  line  cC  drawn  from  the  point 
0,  where  the  LINE  OF  RESISTANCE  abed 
intersects  any  joint  5  6  of  the  struc- 
ture, so  as  to  touch  the  LINE  OF  PRES- 
SURE ABCD,  will  determine  the 
direction  of  the  resultant  pressure 
upon  that  joint:  if  it  lie  within  the  cone  spoken  of,  the 
structure  will  not  slip  upon  that  joint ;  if  it  lie  without  it, 
it  will. 

Thus  the  whole  theory  of  the  equilibrium  of  any  structure 
is  involved  in  the  determination  with  respect  to  that  struc- 
ture of  these  two  lines — the  line  of  resistance,  and  the  line 
of  pressure :  owe  of  these  lines,  the  line  of  resistance,  de- 
termining the  point  of  application  of  the  resultant  of  the 
pressures  upon  each  of  the  surfaces  of  contact  of  the  system  ; 
and  the  other,  the  line  of  pressure,  the  direction  of  that 
resultant. 

The  determination  of  both,  under  their  most  general  forms, 
lies  within  the  resources  of  analysis ;  and  general  equations 
for  their  determination  in  that  case,  in  which  all  the  surfaces 
of  contact,  or  joints,  are  planes — the  only  case  which  offers 
itself  as  &  practical  case — have  been  given  by  the  author  of 
this  work  in  the  sixth  volume  of  the  "  Cambridge  Philo- 
sophical Transactions." 


THE  STABILITY  OF  A  SOLID  BODY. 

285.  The  stability  of  a  solid  body  may  be  considered  to  be 
greater  or  less,  as  a  greater  or  less  amount  of  work  must  be 
aone  upon  it  to  overthrow  it ;  or  according  as  the  amount 


THE   STABILITY    OF   A   STRUCTURE.  381 

of  work  which  must  be  done  upon  it  to  bring  it  into 
that  position  in  which  it  will  fall  over  of  its  own  accord  is 
greater  or  less.  Thus  the  stability  of  the  solid  represented 
in  fig.  1.  resting  on  a  horizontal  Plg  ^  Kjj  2 

plane  is  greater  or  less,  according 
as  the  work  which  must  be  clone 
upon  it,  to  bring  it  into  the  position 
represented  in^g.  2.,  where  its  cen- 
tre of  gravity  is  in  the  vertical 
passing  through  its  point  of  sup- 
port, is  greater  or  less.  Now  this 
work  is  equal  (Art.  60.)  to  that 
which  would  be  necessary  to  raise  its  whole  weight,  verti- 
cally, through  that  height  by  which  its  centre  of  gravity 
is  raised,  in  passing  from  the  one  position  into  the  other. 
Whence  it  follows  that  the  stability  of  a  solid  body  resting 
upon  a  plane  is  greater  or  less,  as  the  product  of  its  weight 
by  the  vertical  height  through  which  its  centre  of  gravity  is 
raised,  when  the  body  is  brought  into  a  position  in  which  it 
will  fall  over  of  its  own  accord,  is  greater  or  less. 

If  the  base  of  the  body  be  a  plane,  and  if  the  vertical 
height  of  its  centre  of  gravity  when  it  rests  upon  a  horizontal 
plane  be  represented  by  A,  and  the  distance  of  the  point  or 
the  edge,  upon  which  it  is  to  be  overthrown,  from  the  point 
where  its  base  is  intersected  by  the  vertical  through  its 
centre  of  gravity,  by  &  ;  then  is  the  height  through  which  its 
centre  of  gravity  is  raised,  when  the  body  is  brought  into  a 
position  in  which  it  will  fall  over,  evidently  represented  by 
(A2  +  #')*—  A;  so  that  if  "W  represent  its  weight,  and  U  the 
work  necessary  to  overthrow  it,  then 

U=W  \(V+lcJ-h\  ....  (376). 
U  is  a  true  measure  of  the  stability  of  the  body. 


THE  STABILITY  OF  A  STRUCTURE. 

286.  It  is  evident  that  the  degree  of  the  stability  of  a 
*  structure,  composed  of  any  number  of  separate  but  contigu- 
ous solid  bodies,  depends  upon  the  less  or  greater  degree  of 
approach  which  the  line  of  resistance  makes  to  the  extrados 
or  external  face  of  the  structure  ;  for  the  structure  cannot  be 
thrown  over  until  the  line  of  resistance  is  so  del.  ected  as  to 


332  rm<:  WALL  OK  PIER. 

intersect  the  extrados :  the  more  remote  is  its  direction  frora 
that  surface,  when  free  from  any  extraordinary  pressure,  the 
less  is  therefore  the  probability  that  any  such  pressure  will 
overthrow  it.  The  nearest  distance  to  which  the  line  of  re- 
sistance approaches  the  extrados  will,  in  the  following  pages, 
be  represented  by  m,  and  will  be  called  the  MODULUS  OF 
STABILITY  of  the  structure. 

This  shortest  distance  presents  itself  in  the  wall  and  but- 
tress commonly  at  the  lowest  section  of  the  structure.  It  is 
evidently  beneath  that  point  where  the  line  of  resistance  in- 
tersects the  lowest  section  of  the  structure  that  the  greatest 
resistance  of  the  foundation  should  be  opposed.  If  that  point 
be  firmly  supported,  no  settlement  of  the  structure  can  take 
place  under  the  influence  of  the  pressures  to  which  it  is  ordi- 
narily subjected.* 


THE  WALL  OR  PIER. 
287.  The  stability  of  a  wall. 

If  the  pressure  upon  a  wall  be  uniformly  distributed  along 
its  length,!  and  if  we  conceive  it  to  be  intersected  by  verti- 
cal planes,  equidistant  from  one  another  and  perpendicular 
to  its  face,  dividing  it  into  separate  portions,  then  are  the 
conditions  of  its  stability,  in  respect  to  the  pressures  applied 
to  its  entire  length,  manifestly  the  same  with  the  conditions 
the  stability  of  each  of  the  individual  portions  into  which  it 
is  thus  divided,  in  respect  to  the  pressures  sustained  by  that 
portion  of  the  wall ;  so  that  if  every  such  columnar  portion 
or  pier  into  which  the  wall  i,s  thus  divided  be  constructed  so 
as  to  stand  under  its  insistent  pressures  with  any  degree  of 
firmness  or  stability,  then  will  the  whole  structure  stand  with 
the  like  degree  of  firmness  or  stability ;  and  conversely. 

In  the  following  discussion  these  equal  divisions  of  the 
length  of  a  wall  or  pier  will  be  conceived  to  be  made  one 
foot  apart ;  so  that  in  every  case  the  question  investigated 
will  be  that  of  the  stability  of  a  column  of  uniform  or  varia- 

*  A  practical  rule  of  Vauban,  generally  adopted  in  fortifications,  brings  the 
point  where  the  line  of  resistance  intersects  the  base  of  the  wall,  to  a  distance 
from  the  vertical  to  its  centre  of  gravity,  of  -|ths  the  distance  from  the  latter 
to  the  external  edge  of  the  base.  (See  Poncelet,  Memoire  sur  la  Stabilite  des 
Hevetemens,  note,  p.  8.) 

f  In  the  wall  of  a  building  the  pressure  of  the  rafters  of  the  roof  is  thua 
uniformly  distributed  by  the  intervention  of  the  wall  plates. 


THE   LINE   OF   RESISTANCE   IN   A   PIER. 


383 


ble  thickness,  whose  width  measured  in  the  direction  of  the 
length  of  the  wall  is  one  foot. 


288.  When  a  wall  is  supported  by  buttresses  placed  at 
equal  distances  apart,  the  conditions  of  the  stability  will  be 
made  to  resolve  themselves  into  those  of  a  continuous  wall, 
if  we  conceive  each  buttress  to  be  ex- 
tended laterally  until  it  meets  the  adja- 
cent buttress,  its  material  at  the  same 
time  so  diminishing  its  specific  gravity 
that  its  weight  when  thus  spread  along 
the  face  of  the  wall  may  remain  the 
same  as  before.  There  will  thus  be  ob- 
tained a  compound  wall  whose  external 
and  internal  portions  are  of  different 
specific  gravities ;  the  conditions  of 
whose  equilibrium  remain  manifestly 
unchanged  by  the  hypothesis  which  has 
been  made  in  respect  to  it. 


THE  LINE  OF  KESISTANCE  IN  A  PIEK. 

289.  Let  ABEF  be  taken  to  repre- 
/''  sent  a  column  of  uniform  dimensions. 

/  Let  PS  be  the  direction  of  any  pres- 

'  &   sure  P  sustained  by  it,  intersecting  its 

axis  in  O.  Draw  any  horizontal  sec- 
tion IK,  and  take  ON  to  represent 
the  weight  of  the  portion  AKIB  of 
the  column,  and  OS  on  the  same  scale 
to  represent  the  pressure  P,  and  com- 
plete the  parallelogram  ONES  ;  then 
will  OK-  evidently  represent,  in  mag- 
nitude and  direction,  the  resultant  of 
the  pressures  upon  the  portion  AKIB 
of  the  mass  (Art.  3.),  and  its  point  of 
intersection  Q  with  IK  will  represent 
a  point  in  the  line  of  resistance. 

Let  PS  intersect  BA  (produced  if  necessary)  in  G,  and  let 
~      ~      "  —  —  ~i\/r/^  "p/~\/"^—  -       *   \-\4-     P 

each  cubic  foot  of  the  material  of  the  mass.     Draw  RL  per- 
pendicular to  CD  ;  then,  by  similar  triangles, 


— 



// 

£ 

I 

IT 

lrn 

• 

1 

«:- 

tt~ 

1 

384:  THE   LINE   OF   KESISTANCE   IN   A   PIEE. 

QM_EL 
OM""OL 


But  QM=y,  OM=CM-CO=z-&  cot._o,  EL=EN 
sin.  ENL=P  sin.  «,  OL=ON+NL=ON+EN  cos.  ENL 
=;«•«»  +P  cos.  a; 

y  P  sin.  a 

"x—k  cot.  a~>#a?H-P  cos.  a  ' 

«  riB.  .-*  COB.  .   ..... 

cos.  a 


which  is  the  general  equation  of  the  line  of  resistance  of  a 
pier  or  wall. 


290.  The  conditions  necessary  that  the  stones  of  the  pier  may 
not  slip  on  one  another. 

Since  in  the  construction  of  the  parallelogram  ONES, 
whose  diagonal  OE  determines  the  direction  of  the  resultant 
pressure  upon  any  section  IK,  the  side  OS,  representing  the 
pressure  P  in  magnitude  and  direction,  remains  always  the 
same,  whatever  may  be  the  position  of  IK  ;  whilst  the  side 
ON,  representing  the  weight  of  AKIB,  increases  as  IK  de- 
scends :  the  angle  EOM  continually  diminishes  as  IK  de- 
scends. Now,  this  angle  is  evidently  equal  to  that  made  by 
OE  with  the  perpendicular  to  IK  at  Q  ;  if,  therefore,  this 
angle  be  less  than  the  limiting  angle  of  resistance  in  the 
highest  position  of  IK,  then  will  it  be  less  in  every  subjacent 
position.  But  in  the  highest  position  of  IK,  ON=0,  so  that 
in  this  position  EOM=a.  Now,  so  long  as  the  inclination 
of  OE  to  the  perpendicular  to  IK  is  less  than  the  limiting 
angle  of  resistance,  the  two  portions  of  the  pier  separated  by 
that  section  cannot  slip  upon  one  another  (Art.  141.).  It  is 
therefore  necessary',  and  sufficient  to  the  condition  that  no 
two  parts  of  the  structure  should  slip  upon  their  common 
surface  of  contact,  that  the  inclination  a  of  P  to  the  vertical 
should  be  less  than  the  limiting  angle  of  resistance  of  the 
common  surfaces  of  the  stones.  All  the  resultant  pressures 
passing  through  the  point  O,  it  is  evident  that  the  line  of 
pressure  (Art.  284.)  resolves  itself  into  that  point. 


THE   LINE   OF   RESISTANCE   IN   A   PIER.  3S5 


291.  The  greatest  height  of  the  pier. 

At  the  point  where  the  line  of  resistance  intersects  the 
external  face  or  extrados  of  the  pier,  y=%a\  if,  therefore,  H 
represents  the  corresponding  value  of  »,  it  will  manifestly 
represent  the  greatest  height  to  which  the  pier  can  be  built, 
so  as  to  stand  under  the  given  insistent  pressure  P.  Substi- 
tuting these  values  for  a?  and  y  in  equation  (377),  and  solving 
in  respect  to  H, 


Psin.  a— 


If  P  sin.  a—  -J-JX&2,  H—  m/£mYy  y  whence  it  follows  that  in 
this  case  the  pier  will  stand  under  the-  given  pressure  P  how- 
ever great  may  be  the  height  to  which  it  is  raised. 


292.  The  line  of  resistance  is  an  equilateral  hyperbola. 

Multiplying  both  sides  of  equation  ($77)  by  the  don  >mi- 
nator  of  the  fraction  in  the  second  member, 

y(pax-\-P  cos.  a)=Px  sin.  a—  P&  cos.  a  ; 

dividing  by  pa,  transposing,  and  changing  the  signs  of  all  the 
terms, 

Psin.  a  Pcos.a         Pcos.a 


/  cos.a\  cos.aT 

—  V  I  #  H  --  1  =  --  —k  : 
M  N»    /          M-a 


adding 


Psin.a/     '  Pcos.a\          /        Pcos.a        PcOS.a/        Psin.a 


/     '       cos.a\          /  cos.a\      PcOS.a/^      Psin.a\ 

\  pa     I     y\  fx«     /         M-a      \  pa     r 

Psin.  a  Pcos.a       Pcos.a  Psin 


en  equa    o  ° 

pa 

VQ=y,,  TV=a),, 


Let  Cil  be  taken  equal  to     ™ifl  nT=—  °—  ;  and  let 

pa  pa 


25 


386  -  THE   LINE   OF   RESISTANCE   IN   A   PIER. 


Pcos.a/7      rsm.a\ 
/.  x,yl= 1  k  + — 1  =  a  constant  quantity. 

This  is  the  equation  of  a  rectangular  hyperbola,  whose 
asymtote  is  TX.*    The  line  of  resist- 
'f       /  ance     continually    approaches    TX 

therefore,  but  never  meets  it ;  whence 
it  follows,  that  if  TX  lie  (as  shown 
in  the  figure)  within  the  surface  of 
the  mass,  or  if  C  H  <  C  B  or 

Psin.ct     .,  013    . 

..  _    .  <£#»  or  2P  sin.  a<f*a.  then 

&''  ^a 

j ,  -'/•  the  line  of  resistance  will  no  where 

cut  the  extrados,  and  the  structure 
will  retain  its  stability  under  the  in- 
sistent pressure  P,  however  high  it 
may  be  built ;  which  agrees  with 
the  result  obtained  in  the  preceding 
article. 


ii/frf 


293.  The  thickness  of  the  pier,  so  that  when  raised  to  a  given 
height  it  may  have  a  given  stability. 

Let  m  be  taken  to  represent  the  nearest  distance  to  which 
the  line  of  resistance  is  intended  to  approach  the  extrados  of 
the  pier,  which  distance  determines  the  degree  of  its  stability, 
and  has  been  called  the  modulus  of  stability  (Art.  286.).  It 
is  evident  from  the  last  article  that  this  least  distance  will 
present  itself  in  the  lowest  section  of  the  pier.  At  this 
lowest  section,  therefore,  y=^a—m.  Substituting  this  value 
for  y  in  equation  (377),  and  also  the  height  h  of  the  pier  for 
cc,  and  solving  the  resulting  quadratic  equation  in  respect  to 
a,  we  shall  thus  obtain 

rP  cos.  a 


2P 
fr 

Church's  Analyt.  Geom.,  Art.  161. 


\  { 

)cos.a[..(379). 


A   WALL    SUPPORTED   BY    SHORES. 


387 


294.  To  vary  the  point  of  application  of  the  pressure  P,  so 
that  any  required  stability  may  he  given  to  the  pier. 

It  is  evident,  that  if  in  equation  (377)  we  substitute  \a,—m 
for  y,  and  k  for  a?,  the  modulus  of  stability  m 
may  be  made  to  assume  any  given  value  for 
a  given  thickness  a  of  the  pier,  by  assigning 
a  corresponding  value  to  k ;  that  is,  by  mov- 
ing the  point  of  application  G-  to  a  certain 
distance  from  the  axis  of  the  pier,  deter- 
mined by  the  value  of  Jc  in  that  equation. 
This  may  be  done  by  various  expedients,- 
and  among  others  by  that  shown  in  the 
figure.  Solving  equation  (377)  in  respect  to 
&  we  have 

. . . .  (380), 

It  is  necessary  to  the  equilibrium  of  the  pier,  under  these 
circumstances,  that  the  line  of  resistance  should  no  where 
intersect  its  intrados  below  the  point  D. 


\ 


THE  STABILITY  OF  A  WALL  SUPPORTED  BY  SHORES. 

295.  Let  the  weight  of  the  portion  of  the  wall  supported 
by  each  shore  or  prop,  and  the 
>T?  pressure  insistent  upon  it,  be  im- 
/  agined  to  be  collected  in  a  single 
foot  of  the  length  of  the  wall ;  tho 
conditions  of  the  stability  of  tho 
wall  evidently  remain  unchanged 
by  this  hypothesis.  Let  ABCD 
represent  one  of  the  columns  or 
piers  into  which  the  wall  will  thus 
be  divided,  EF  the  corresponding 
shore,  P  the  pressure  sustained  upon 
the  summit  of  the  wall,  Q  the 
thrust  upon  the  shore  EF,  2w  its 
weight,  x  the  point,  where  the  line 
of  resistance  intersects  the  base  of 
the  wall,  Csc=m,  CJ?=b,  FEC=£: 
and  let  the  same  notation  be  taken  in  other  respects  as  in 


388  A   WALL    SUPPORTED   BY   SHORES. 

the  preceding  articles.  Then,  since  a?  is  a  point  in  the  direc 
tion  of  the  resultant  of  the  resistances  by  which  the  base  of 
the  column  is  sustained,  the  sum  of  the  moments  about  that 
point  of  the  pressure  P  and  half  the  weight  of  the  shore, 
supposed  to  be  placed  at  E*,  is  equal  to  the  sum  of  the 
moments  of  the  thrust  Q,  and  the  weight  pah  of  the  column; 
or  drawing  soM.  and  %N  perpendiculars  upon  the  directions 
of  P  and  Q, 

P. 

Now  #M  =  xs  sin.  a?*M=(HK—  Ht)  sin.  a  —  ^— 
cot.  a}  sin.  a=A  sin.  a—  (k+^a—  mj  cos.  a,  %~N=(b+m)  cos. 


0—  m)cos.  aj  + 
wm=Q(b+m)  cos.  P  +  pahQa—  m) 


Solving  this  equation  in  respect  to  Q,  and  reducing,  we 
obtain. 


+"*  cos. 

This  expression  may  be  placed  under  the  form 
Q=(P  cos.  a+pah+w)  sec.  (3— 
Pjfrcos.  a—  A  sin,  a  +  (&+-|#)  cos,  a 


If  the  numerator  of  the  fraction  in  the  second  member  of 
this  equation  be  a  positive  quantity  (as  in  all  practical  cases 
it  will  probably  be  found  to  be)  the  value  of  Q  manifestly 
diminishes  with  that  of  m.  Now  the  least  value  of  m,  con- 
sistent with  the  stability  of  the  wall,  is  zero,  since  the  line 
of  resistance  no  where  intersects  the  extrados;  the  least 
value  of  Q  (the  shore  being  supposed  necessary  to  the  sup- 
port of  the  wall)  corresponds,  therefore,  to  the  value  zero  of 
m  ;  moreover  this  least  value  of  the  thrust  upon  the  shore 
consistent  with  the  stability  of  the  wall  is  manifestly  that 
which  it  sustains  when  the  wall  simply  rests  upon  it,  the 

*  The  weight  2w  of  the  shore  may  be  conceived  to  be  divided  into  two  equal 
parts  and  collected  at  its  extremities. 

•}•  The  expression  (b-\-m)  cos.  ft  may  be  placed  under  the  form  b  cot  ft  sin. 
3-if-m  cos.  0=c  sin.  ft-\-m  cos.  3,  where  c  represents  the  height  CE  of  the  point 
against  which  the  prop  rests. 


WALL  SUPPORTED  BY  SHORES. 


389 


shore  not  being  driven  so  as  to  increase  the  thrust  sustained 
by  it  Beyond  that  just  necessary  to  support  the  wall.* 
This  least  thrust  is  represented  by  the  formula 


rt__FjAsm.q— 

" 


cos.  a  — 


The  thrust  which  must  be  given  to  the  prop  in  order  that 
there  may  be  given  to  the  wall  any  required  stability,  deter- 
mined by  the  arbitrary  constant  m,  is  determined  by  equa- 
tion (381).  The  stability  will  diminish  as  the  value  of  m  is 
increased  beyond  £<z,  and  the  wall  will  be  overthrown 
inwards  when  it  exceeds  a. 


296.  The  stability  of  a  wall  sustained  lyy  more  than  one 
shore  in  the  same  plane. 

Let  EF,  ef  be  shores  in  the  same  plane,  sustaining  the 
wall  ABCD,  and  both  necessary  to 
its  stability;  so  that  if  EF  were  re- 
moved, the  wall  would  turn  over  upon 
/",  and  if  ef  were  removed,  upon  some 
point  between  F  and  C. 

If  the  thrust  of  the  shore  EF  be 
only  that  just  necessary  to  sustain 
the  tendency  of  the  wall  to  overturn 
upon  /,  it  is  evident  that  the  line  of 
resistance  must  pass  through  that 
point ;  but  if  the  thrust  exceed  that 
just  necessary  to  the  equilibrium,  or 
if  the  shore  be  driven  then  the  line 
of  resistance  will  intersect  fg  in  some 
points.  ~Letfx=m;  then  represent- 
ing the  thrust  upon  EF  by  Q,  the  dis- 
tances f  D  and  fi  by  h  and  &,  and  the  angle  EFC  by  /3,  the 
value  of  Q  is  evidently  determined  by  equation  (381). 

If  z  be  taken  in  like  manner  to  represent  the  point  where 
the  line  of  resistance  intersects  the  base  of  the  wall,  and 
Cz=ml,  CE— b, ;  Ce=b»  Cfe^P^  GD= A,,  the  thrust  upon 
the  prop  ef  by  Qt  and  its  weight  by  2wl ;  then  the  sum  of 
the  moments  about  the  point  z  of  Q  and  Q1?  and  the  weight 


*  This  case  presents  an  application  of  the  principle  of  least  resistance. 
Theory  of  the  Arch.) 


(See 


390 


THE    STABILITY   OF   A   GOTHIC    STRUCTURE. 


J  of  the  wall,  equals  the  sum  of  the  moments  of  P,  w, 
and  i0j  ;  or 

QX&i+ff^)  cos.  ft  +  Q  (#1+^1)  cos.  (3+pahi  (%a—m^)= 
P{A!  sin.  a—  (&-fia—  m^  cos.  a}'  +  (^0+^)  m,  ......  (382.) 

Substituting  the  value  of  Q  in  this  equation,  from  equation 
(381),  and  solving  in  respect  to  Q,,  the  thrust  upon  the  prop 
cf  will  be  determined,  so  that  the  stability  of  the  wall,  upon 
its  section  fg  and  upon  its  base  CB,  may  be  m  and  mt 
respectively. 

If  m1=mt  the  portions  of  the  wall  above  and  below  fg 
are  equally  stable. 

If  m1=m=Q,  the  thrust  upon  each  shore  is  only  that 
which  is  just  necessary  to  support  the  wall,  or  which  is  pro- 
duced by  its  actual  tendency  to  overturn.  In  this  case  we 
have 


(P  sin,  a-^a3)  foft- 


ft) 


cos,  a 


~  bbt  cos.  ft 

the  value  of  k  being  determined  by  equation  (380). 


Let 


297.  The  stability  of  a  structure  having  parallel  walls,  one 
of  which  is  supported  l)y  means  of  struts  resting  on  the 
summit  of  the  other. 

AB  and  CD  be  taken  to  represent  the  walls,  and  EF 
one  of  the  struts ;  the  thrust  Q  upon  the 
strut  may  be  determined  precisely  as  in 
Art  295.  So  that  the  line  of  resistance 
may  intersect  the  base  of  the  wall  AB  at 
a  given  distance  m  from  the  extrados 
(see  note,  p.  388.) 

Let  m,  represent  the  distance  Dx  from 
the  extrados  at  which  the  line  of  resist- 
ance intersects  the  base  of  the  wall  CD ; 
then  taking  the  moments  of  the  pressures 
applied  to  the  wall  CD  about  the  point 
a?,  as  in  Art.  295,  and  observing  that 
besides  the  pressure  Q  the  weight  w  of 
one  half  the  strut  is  applied  at  E,  we 
have 

X  sin. 


THE    STABILITY   OF   A   GOTHIC    STKUOTITRE.  391 

in  which  equation  h^  and  at  are  taken  to  represent  the 
height  and  thickness  of  the  wall  CD,  \  the  distance  of  the 
point  E  on  which  the  strut  rests  from  the  axis  of  the  wall,  (3 
the  inclination  of  the  strut  to  the  vertical,  and  ^  the  weight 
of  a  cubic  foot  of  the  material  of  the  wall. 

Substituting  for  Q  its  value  from   equation   (381),  and 
reducing, 


P{Asin.«—  (k+%a)  cos.  aj  —  ^c?h+m(F  cos,  a  -f 
c  sin.  ft+m  cos.  ft 


sin.  ft 


/OOON 


By  this  equation  is  determined  that  relation  between  the 
dimensions  of  the  two  walls  and  the  amount  of  the  insistent 
pressure  P,  by  which  any  required  stability  may  be  assigned 
to  each  wall  of  the  structure.  If  m=0,  the  pressure  upon 
the  strut  will  be  that  only  which  is  produced  by  the  ten- 
dency of  AB  to  overturn  ;  and  the  value  of  m^  determined 
from  the  above  equation  will  give  the  stability  of  the  exter- 
nal wall  on  this  supposition. 

*  If  m=0  and  m1=0,  both  walls  will  be  upon  the  point  of 
overturning,  and  the  above  equation  will  express  that  rela- 
tion between  the  dimensions  of  the  wall  and  the  amount  of 
the  insistent  pressure,  which  corresponds  to  the  state  of  the 
instability  of  the  structure. 

The  conditions  of  the  stability,  when  the  wall  AB  is  sup- 
ported by  two  struts  resting  upon  the  summit  of  the  wall 
CD,  may  be  determined  by  a  method  similar  to  the  above 
(see  Art.  296). 

The  general  conditions  of  the  stability  of  the  structure 
discussed  in  this  article  evidently  include  those  of  a  GOTHIC 
BUILDING  having  a  central  nave,  whose  walls  are  supported, 
under  the  thrust  of  its  roof,  by  the  rafters  of  the  roof  of  its 
side  aisles.  By  a  reference  to  the  principles  of  the  preceding 
article,  the  discussion  may  readily  be  made  to  include  the 
case  in  which  a  further  support  is  given  to  the  walls  of  tho 
nave  by  flying  buttresses,  which  spring  from  the  summits  of 
the  walls  of  the  aisles.  The  influence  of  the  buttresses 
which  support  the  walls  of  the  aisles  upon  the  conditions  of 
the  stabiiity  of  the  structure  forms  the  subject  of  a  subse- 
quent article. 


392 


THE   WALL   OF  A  DWELLING. 


298.  The  stability  of  a  wall  sustaining  the  floors  of  a 
dwelling. 


The 


joists  of  the  floors  of  a  dwelling-house  rest  at  their 
extremities  upon,  and  are  sometimes 
notched  into,  pieces  of  timber  called 
wall-plates,  which  are  imbedded  in 
the  masonry  of  the  wall.  They 
serve  thus  to  bind  the  opposite  sides 
of  the  house  together ;  and  it  is  upon 
the  support  which  the  thin  walls  of 
modern  houses  receive  from  these 
joists,  that  their  stability  is  some- 
times made  to  depend.* 

Representing 
that  portion  oi 

rests  upon  the  portion  ABCD  of  the 
wall,  and  the  distance  BE  by  <?, 
taking  a?,  as  before,  to  represent  the 
point  where  the  line  of  resistance 
intersects  the  base  of  the  wall,  and 
measuring  the  moments  from  thig 


by  w  the  weight  of 
the  flooring  which 


point,  we  have 


whence,  taking  the  same  notation  as  in  the  preceding  arti- 
cles, and  substituting, 

<)Q  +  (%a—m)pah+(a—m)w=  {  Asm.  a—  (k+^a—  m)  cos.  a}  P  ; 
,VQc  =  \h  sin.  a—  (k+$a)  cos.  a}  P—  J^«2A— 
m  (P  cos.  a+iiah+w)  .....  (384)  ; 


from  which  expression  it  appears  that  Q  is  less  as  m  is  less. 
"When,  therefore,  the  strain  upon  the  joints  is  that  only 
which  is  just  necessary  to  preserve  the  stability  of  the  wall, 
or  which  it  produces  by  its  tendency  to  overturn,  then 
m=0.  In  this  case,  therefore, 


*  A  house  thus  constructed  evidently  becomes  unsafe  when  its  wall-plates  or 
the  extremities  of  its  joists  begin  to  decay. 


A   WALL    SUPPORTED   BY   BUTTRESSES.  393 

\h  Sill,  a  —  (&-J--J-&)  COS.  a}  P  — \\L(ji?h~W(L  /oosrs 

2=  -  G  ~ (doo). 

If  13  be  assumed  a  right  angle,  and  if  (a—m)w  be  substi- 
tuted for  mw,  the  case  discussed  in  Art.  295.  will 
evidently  pass  into  that  which  is  the  subject  of 
the  present  article,  and  the  preceding  equation 
may  thus  be  deduced  from  equation  (381)  (see 
note,  p.  388.). 

In  like  manner,  if  the  wall  sustain  the  pres- 
sure of  two  floors,  and  A  be  taken  to  represent 
the  distance  from  its  summit  to  the  lower  floor, 
and  AJ  its  whole  height ;  then,  representing  by  m 
arid  rax  the  distances  from  the  extrados  at  which 
the  line  of  resistance  intersects  the  sections  EG 
and  eg,  and  substituting  (w  +  w^)  (a  —  m^  for 
(w +w1)m1,  the  value  of  the  strain  Q  on  the 
joists  of  the  lower  floor  may  be  determined  by 
equation  (382),  it  being  observed  that  for  the 
coefficient  of  Qt  in  that  equation  must  be  substi- 
tuted (as  was  shown  above)  the  height  (hl—h]  of 
the  lower  floor  from  the  bottom  of  the  wall.  If  the  strain 
be  only  that  produced  by  the  tendency  of  the  wall 
turn  at  g  and  C,  then 


W  Cd 

.  0,+wa— 


—  P  sin.  a 


The  value  of  Q  is  determined  by  equation  (385),  G  being 
taken  to  represent  the  distance  E0  between  the  floors.  If 
the  joists  be  not  notched  into  the  wall-plates,  the  friction  of 
their  extremities  upon  them,  produced  per  foot  of  the  length 
by  the  weight  which  they  support,  must  at  least  equal  Q  and 
Qi  respectively. 


299.  The  stability  of  a  wall  supported  by  piers  or  buttresses 
of  uniform  thickness. 

Let  the  piers  be  imagined  to  extend  along  the  whole 


394:  WALL    SUPPORTED   BY   BUTTRESSES. 


length  of  the  wall,  as  explained  in  Art.  288.  ; 
and  let  ABCD  represent  a  section  of  the  com- 
pound wall  thus  produced.  Let  the  weight  of 
each  cubic  foot  of  the  material  of  the  portion 
ABFE  be  represented  by  f*1?  and  that  of  each 
cubic  foot  of  GFCD  by  f*f,  EA=«1,  GD  =  a» 
BC=a,  AB  =  A15  CD  =  A2,  distance  from  CD, 
produced,  of  the  point  where  P  intersects 
AE=Z,  x  the  intersection  of  the  line  of  resist- 
ance with  CB,  Cx=  m.  By  the  principle  of  the 
equality  of  moments,  the  moment  of  P  about 
the  point  x  is  equal  to  the  sum  of  the  moments 
of  the  weights  of  GO  and  AF  about  that  point. 
But  (Art.  295.)  moment  of  P=P  \h,  sin.  a— 
(l—m)  cos.  aj  ;  also  moment  of  weight  of  AF— 
(az—m  +-J#1)A1&1M-1  ;  moment  of  weight  of  GC= 


P  {At  sin.  a—  (l—m)  cos.  *}  =(»,— 

.....  (387) 


If  the  material  of  the  pier  be  the  same  with  that  of  the 
wall  ;  then,  taking  b  to  represent  the  breadth  of  each  pier, 
and  c  the  common  distance  of  the  piers  from  centre  to 
centre  (Art.  288.),  ca^=ba^  therefore  c^=b^l.  Repre- 

si 

senting  =-  by  n,  eliminating  the  value  of  ^  between  this 
equation  and  equation  (387),  writing  ^  for  i^,  and  reducing, 
^  sin.  a—  I  cos.  «)=if*  (afh^  +  20,0^  +  -«22A3  1  — 

\  7l>  I 


ra-JPcos. a+M< \alhl+-ajit]  \.  .  .  .(388); 
/  \  n      V  \ 

by  which  equation  a  relation  is  determined  between  the 
dimensions  of  a  wall  supported  by  piers,  having  a  given 
stability  m,  and  its  insistent  pressure  P.  Solving  it  in 
respect  to  &2,  the  thickness  of  the  pier  necessary  to  give  any 
required  stability  to  the  wall  will  be  determined.  (See 
APPENDIX.) 

If  #2  be  assumed  to  represent  that  width  of  the  pier  by 
which  the  wall  would  just  be  made  to  sustain  the  given 
pressure  P  without  being  overthrown;  then  taking  ra=0, 
and  solving  in  respect  to  #a, 


WALL   SUPPORTED   BY   BUTTRESSES.  395 


sin.  «-Z  cos.  «)+        (       -1))  *•„....  (389). 
Aa    \  Aa  / 


300.    .7^0  stability  of  a  pier  or  buttress  swr* 
mounted  by  a  pinnacle. 

Let  "W  represent  the  weight  of  the  pinnacle, 
and  e  the  distance  of  a  vertical  through  its  cen- 
tre of  gravity  from  the  edge  0  of  the  pier :  then 
assuming  x  to  be  the  point  where  the  line  of 
resistance  intersects  the  base  of  the  pier,  and  tak- 
ing the  same  notation  as  before,  equation  (387) 
will  evidently  become 


P  {k,  sin.  »—(l—m)  cos.  *}  =  \a%— 

+  (e—m)W. 


Substituting  for /xa  its  value-^  or— ,   writing    p  for  f^,   and 
reducing, 

P(A!  sin.  a — I  cos.  *)=if*|a1>A1-f2fl1ofA1-ht-al'Atj  + 


If  a9  represent  the  thickness  of  that  pier  by  which  the  wall 
will  just  be  sustained  under  the  pressure,  taking  m=0,  and 

solving  in  respect  to  aa,  a^=—nal-^-  + 


fP(A,  Bin.  .-I  cos.  .)_-W«|  +  '-l  «,'  •  •  (391). 


396  WATT.   SUPPORTED   BY   GOTHIC  BUTTRESSES. 


THE  GOTHIC  BUTTRESS. 

301.  In  Gothic  buildings  the  thickness  of 
a  buttress  is  not  unfrequently  made  to  vary 
at  two  or  three  different  heights  above  its 
base.  Such  buttress  is  represented  in  the 
accompanying  figure. 

The  conditions  by  which  any  required  sta- 
bility may  be  assigned  to  that  portion  of  it 
whose  base  is  ~be  may  evidently  be  determined 
by  equation  (390).  To  determine  the  condi- 
tions of  the  stability  of  the  whole  buttress 
upon  CD,  let  the  heights  of  the  points  Q,  a, 
and  b  above  CD  be  represented  by  A,,  A2  and 
7*3;  let  DE=^,  DY=a»  FC=a3,  Cx=m1; 
then  adopting,  in  other  respects,  the  same 
notations  as  in  Arts.  299  and  300.  Since  the 
distances  from  x  of  the  verticals  through  the 
centres  of  gravity  of  those  portions  of  the 
buttress  whose  bases  are  DE,  DF,  and  FC 
respectively,  are  (a^  +  a^^a.—m^  (0,-fi0a 
_ m^  and  (:kas—'m1)  we  have,  by  the  equality 
of  moments, 

P  {AJ  sin.  a— (I— m,)  cos.  a}  =(^  +  ^4-^— m,)  AAM-  + 

(^-m1) (392). 


This  equation  establishes  a  relation  between  the  dimen- 
sions of  the  buttress  and  its  stability,  by  which  any  one  of 
those  dimensions  which  enter  into  it  may  be  so  determined 
as  to  give  to  mt  any  required  value,  and  to  the  structure  any 
required  degree  of  stability.  (See  APPENDIX.) 

It  is  evident  that,  with  a  view  to  the  greatest  economy  of 
the  material  consistent  with  the  given  stability  of  the  but- 
tress, the  stability  of  the  portion  which  rests  upon  the  base 
be  should  equal  that  of  the  whole  buttress  upon  CE ;  the 
value  of  m^  in  the  preceding  equation  should  therefore  equal 
that  of  m  in  equation  (390)  If  m  be  eliminated  between 
these  two  equations,  it  being  observed  that  hl  and  A2  in  equa> 
tion  (390)  are  represented  by  A,—  A3  and  A2— A3  in  equation 
(392),  a  relation  will  be  established  between  a1?  #„  a3  A1?  A,, 
A8,  which  relation  is  necessary  to  the  greatest  economy  of 


THE   STABILITY   OF   WALLS   SUSTAINING   KOOFS.  397 

material ;  and  therefore  to  the  greatest  stability  of  the  struc- 
ture with  a  given  quantity  of  material. 


THE  STABILITY  OF  WALLS  SUSTAINING  ROOFS. 

302.  Thrust  upon  the  feet  of  the  rafters  of  a  roof,  the  tie- 
beam  not  being  suspended  from  the,  ridge. 

If  f^  be  taken  to  represent  the  weight  of  each  square  foot 
of  the  roofing,  2L  the  span,  i  the 
inclination  BAG  of  the  rafters  to 
the  horizon,  q  the  distance  between 
each  two  principal  rafters,  and  a 


h1     the  inclination  to  the  vertical  of 
the  resultant    pressure   P   on  the 

foot  of  each  rafter  ;  then  will  L  sec.  i  represent  the  length  of 
each  rafter,  and  f^L^  sec.  i  the  weight  of  roofing  borne  by 
each  rafter.  Let  the  weights  thus  borne  by  each  of  the 
rafters  AB  and  BC  be  imagined  to  be  collected  in  two  equal 
weights  at  its  extremities  ;  the  conditions  of  the  equilibrium 
will  remain  unchanged,  and  there  will  be  collected  at  B  the 
weight  supported  by  one  rafter  and  represented  by  f^L^ 
sec.  t,  and  at  A  and  C  weights,  each  of  which  is  represented 
by  -J^L^  sec.  i.  Now,  if  Q  be  taken  to  represent  the  thrust 
produced  in  the  direction  of  the  length  of  either  of  the 
rafters  AB  and  BC,  then  (Art.  13.)  ^Lq  sec.  *  =  2Q  cos. 
|ABC:  but  ABC^tf  —  2t;  therefore  cos.  JABC  =  sin.  *; 
therefore  2Q  sin.  i=p1~Loi  sec.  i  ; 

.  o-    T  .8ec'*  M^Lg  f^Lg 

•'•  ""^V^!  sin.  L  ~~  2  sin.  i  cos.  i  ~~  sin.  2*' 

The  pressures  applied  to  the  foot  A  of  the  rafter  are  the 
thrust  Q  and  the  weight  -J^L^  sec.  i  ;  and  the  required  pres- 
sure P  is  the  resultant  of  these  two  pressures.  Resolving  Q 
vertically  and  horizontally,  we  obtain  Q  sin.  i  and  Q  cos.  «, 
or  ^^Lq  sec.  L  and  j^L^  cosec.  i.  The  whole  pressure  applied 
vertically  at  A  is  therefore  represented  by  p^Lq  sec.  t,  and 
the  whole  horizontal  pressure  by  ^^Lq  cosec.  i  ;  whence  it 
follows  (Art.  11.)  that 


P  =  V'        sec.  1*+Jf*I'L     cosec.  '<= 


i  t/l+icot.2*  .....  (393). 


398  RAFTERS   OF  A  ROOF. 


(394)" 


If  the  inclination  i  of  the  roof  be  made  to  vary,  the  span 
remaining  the  same,  P  will  attain  a  minimum  value  when 

tan.  i  =  —  -,  or  when 

4/2 

£=35°  16'  .....  (395). 

It  is  therefore  at  this  inclination  of  the  roof  of  a  given 
span,  whose  trusses  are  of  the  simple  form  shown  in  the 
figure,  that  the  least  pressure  will  be  produced  upon  the  feet 
of  the  rafters.  If  <p  represent  the  limiting  angle  of  resistance 
between  the  feet  of  the  rafters  and  the  surface  of  the  tie,  the 
feet  of  the  rafters  would  not  slip  even  if  there  were  no  mor- 
tice or  notch,  provided  that  a  were  not  greater  than  <p  (Art. 
141.),  or  \  cot.  i  not  greater  than  tan.  9,  or 

i  not  less  than  cot.-1  (2  tan.  ?>)*  .....  (396). 


303.  The  thrust  upon  the  feet  of  the  rafters  of  a  roof  in 
which  the  tie-beam  is  suspended  from  the  ridge  l>y  a 
king-post. 

It  will  be  shown  in  a  subsequent  portion  of  this  work 
(see  equation  558)  that,  in  this  case, 
the  strain  upon  the  king-post  BD  is 
equal  to  fths  of  the  weight  of  the 
tie-beam  with  its  load.  Represent- 
ing,  therefore,  the  weight  of  each 
foot  in  the  length  of  the  tie-beam 
by  f*a,  and  proceeding  exactly  as  in 
the  last  article,  we  shall  obtain  for  the  pressure  P  upon  the 
feet  of  the  rafters,  and  its  inclination  to  the  vertical,  the 
expressions 

+  5M'2)2cot.2^^  ----  (397). 


'  .  .(398). 

- 


*  If  the  surfaces  of  contact  be  oak,  and  thin  slips  of  oak  plank  be  fixed 
under  the  feet  of  the  rafters,  so  that  the  surfaces  of  contact  may  present  par- 
allel fibres  of  the  wood  to  one  another  (by  which  arrangement  the  friction  will 
be  greatly  increased),  tan.  <j>=-48  (see  p.  133.);  whence  it  follows  that  the 
rafters  will  not  slip,  provided  that  their  inclination  exceed  cot."1  -96,  or 
46°  10'. 


WALL    SUSTAINING   THE   THRUST    OF   A   KOOF. 


399 


304.  The  stability  of  a  wall  sustaining  the  thrust  of  a  roof, 
having  no  tie-beam. 


Let 


it  be  observed,  that  in  the  equation  to  the  line  of 
resistance  of  a  wall  (equation 
377),  the  terms  P  sin.  a  and  P 
cos.  a  represent  the  horizontal 
and  vertical  pressures  on  each 
foot  of  the  length  of  the  summit 
of  the  wall  ;  arid  that  the  former 
of  these  pressures  is  represented 
in  the  case  of  a  roof  (Art.  302.) 
by  i^L  cosec.  £,  and  the  latter 
by  M-jL  sec.  i  ;  whence,  substitu- 
ting these  values  in  equation 
(377),  we  obtain  for  the  equation 
to  the  line  of  resistance  in  a  wall 
sustaining  the  pressure  of  a  roof, 
without  a  tie-beam 


I— I 

H*!/ 


_ 


in  which  expression  a  represents  the  thickness  of  the  wall, 
k  the  distance  of  the  feet  of  the  rafters  from  the  centre  of 
the  summit  of  the  wall,  L  the  span  of  the  roof,  ^  the  weight 
of  a  cubic  foot  of  the  wall,  and  /*,  the  weight  of  each  square 
foot  of  the  roofing.  The  thickness  a  of  the  wall,  so  that, 
being  of  a  given  height  A,  it  may  sustain  the  thrust  of  a 
roof  of  given  dimensions  with  any  given  degree  of  stability, 
may  be  determined  precisely,  as  in  Art.  293,  by  substituting 
h  for  x  in  the  above  equation,  and  \a— m  for  y,  and  solving 
the  resulting  quadratic  equation  in  respect  to  a. 

If,  on  the  other  hand,  it  be  required  to  determine  what 
must  be  the  inclination  i  of  the  rafters  of  the  roof,  so  that 
being  of  a  given  span  L  it  may  be  supported  with  a  given 
degree  of  stability  by  w^alls  of  a  given  height  h  and  thick- 
ness a/  then  the  same  substitutions  being  made  as  before, 
the  resulting  equation  must  be  solved  in  respect  to  i  instead 
of  a. 

The  value  of  a  admits  of  a  minimum  in  respect  to  the 
variable  i.  The  value  of  t,  which  determines  such  a  mini- 
mum value  of  a,  is  that  inclination  of  the  rafters  which  is 


4:00  STABILITY   OF   A   WALL. 

consistent  with  the  greatest  economy  in  the  material  of  the 
wall,  its  stability  being  given. 


305.    The  stability  of  a  wall  supported  ~by  buttresses,  and 
sustaining  the  pressure  of  a  roof  without  a  tie-beam. 

The  conditions  of  the  stability  of  such  a  wall,  when  sup- 
ported by  buttresses  of  uniform  thickness,  will  evidently  be 
determined,  if  in  equation  (388)  we  substitute  for  P  cos.  a 
and  P  sin.  a  their  values  f^L  sec.  i  and  -J^L  consec.  i ;  we 
shall  thus  obtain 

cosec.  i— I  sec.  *)=Jf*  (01iA1+2alo8M — a?h^—m 

n 

fsL  sec.  I+P  (#A  +  -  <*A  f (400). 

From  which  equation  the  thickness  az  of  the  buttresses 
necessary  to  give  any  required  stability  m  to  the  wall  may 
b#  determined. 

If  the  thickness  of  the  buttresses  be  different  at  different 
heights,  and  they  be  surmounted  by  pinnacles,  the  con- 
ditions of  the  stability  are  similarly  determined  by  substi- 
tuting for  P  sin.  a  and  P  cos.  a  the  same  values  in  equations 
(390)  and  (392). 

To  determine  the  conditions  of  the  stability  of  a  Gothic 
building,  whose  nave,  having  a  roof  without  a  tie-beam,  is 
supported  by  the  rafters  of  its  two  aisles,  or  by  flying  but- 
tresses, which  rest  upon  the  summits  of  the  walls  of  its 
aisles,  a  similar  substitution  must  be  made  in  equation  (383). 

If  the  walls  of  the  aisles  be  supported  by  buttresses, 
equation  (383)  must  be  replaced  by  a  similar  relation 
obtained  by  the  methods  laid  down  in  Arts.  299  and  301 ; 
the  same  substitution  for  P  sin.  a  and  P  cos.  a  must  then  be 
made. 


306.  The  conditions  of  the  stability  of  a  wall  supporting  a 

shed  roof. 

Let  AB  represent  one  of  the  rafters  of  such  a  roof,  one  ex- 


STABILITY   OF   A   WALL. 


401 


tremity  A  resting  against  the  face  of 
the  wall  of  a  building  contiguous  to 
the  shed,  and  the  other  B  upon  the 
r   summit  of  the  wall  of  the  shed. 

It  is  evident  that  when  the  wall 
BH  is  upon  the  point  of  being  over- 
thrown, the  extremity  A  will  be  upon 
the  point  of  slipping  on  the  face  of 
the  wall  DC ;  so  that  in  this  state  of 
the  stability  of  the  wall  BH,  the  direc- 
tion of  the  resistance  K  of  the  wall 
DC  on  the  extremity  A  of  the  rafter, 
will  be  inclined  to  the  perpendicular  AE  to  its  surface  at.ani 
angle  equal  to  the  limiting  angle  of  resistance.  Moreover;, 
this  direction  of  the  resistance  R  which  corresponds  to  the- 
state  bordering  upon  motion  is  common  to  every  other  state  ; 
for  by  the  principle  of  least  resistance  (see  Theory  of  the 
Arch)  of  all  the  pressures  which  might  be  supplied  by  the 
resistance  of  the  wall  so  as  to  support  the  extremity  of  the 
rafter,  its  actual  resistance  is  the  least..  Now  this  least  re- 
sistance is  evidently  that  whose  direction  is  most  nearly  ver- 
tical ;  for  the  pressure  upon  the  rafter  is  wholly  a  vertical 
pressure.  But  the  surface  of  the  wall  supplies  no  resistance 
whose  direction  is  inclined  farther  from  the  horizontal  line 
AE  than  AR  ;  AR  is  therefore  the  direction  of  the  resist- 
ance. 

Resolving  R  vertically  and  horizontally,  it  becomes  R  sin. 
9  and  R  cos.  9.  Representing  the  span  BF  by  Lv  the  incli- 
nation ABE  by  i,  the  distance  of  the  rafters  by  £,  and  the 
weight  of  each  square  foot  of  roofing  by  f\  (Art.  10.),  R  sin. 
9  +  P  cos.  a,=t*<l  Lq  sec.  i  and  Reos.  9— P  sin.  a=0 ;  also  the 
perpendiculars  let  fall  from  A  on  P  and  upon  the  vertical 
through  the  centre  of  AB,  are  represented  by 

L  cos.  (a,  +  i)  sec.  i  and  -JL ;  therefore  (Art.  7). 
PL  cos.  (a  +  *)sec.  i=^L  .  L^  q  sec.  t,  and  hence 
P  cos.  (a  +  i)=^'L^l  q.     Eliminating  between  these  equa- 
tions, we  obtain 


cot.  a=tan.  9  +  2  tan.  t 


(401); 


sin.(9  +  0' 


cos.  t(  tan.  9  +  tan.  i) 
26 


402 


THE   PLATE   BANDE. 


If  the  rafter,  instead  of  resting  at  A 
against  the  face  of  the  wall,  be  received 
into  an  aperture,  as  shown  in  the  figure, 
so  that  the  resistance  of  the  wall  may  be 
applied  upon  its  inferior  suface  instead  of 
at  its  extremity:  then  drawing  AE  per- 
pendicular to  the  surface  of  the  rafter, 
the  direction  AR  of  the  resistance  is  evi- 
dently inclined  to  that  line  at  the  given 
limiting  angle  9.  Its  inclination  to  the  hori- 
zon is  therefore  represented  by  -^— 

Substituting  this  angle  for  9  in  equations  (401)  and  (402), 

cot.  a=cot.  (*— 9)  +  2  tan.  i (403). 

,2  sec.  i 


cos.  (i— 


.  (i— 9)tan.  i ' 


cos.t{cot.(t-9)+tan.i 


. 


Substituting  in  equations  (3  77)  and  (379)  for  Psin.  a,  P  cos.  a, 
their  values  determined  above,  all  the  conditions  of  the  sta- 
bility of  a  wall  supporting  such  a  roof  will  be  determined. 


307.  THE  PLATE  BANDE  OR  STRAIGHT  ARCH. 


Let  MN  represent  any  joint  of 
the  plate  bande  ABCD,  whose 
points  of  support  are  A  and  B  ; 
PA  the  direction  of  the  resistance 
at  A,  WQ  a  vertical  through  the 
centre  of  gravity  of  AMND,  TR 
the  direction  of  the  resultant  pres- 
sure upon  M.'N  ;  the  directions  of 

TR,  WQ,  and  PA  intersect,  therefore,  in  the  same  point  O. 
Let  OAD=a,  AM=aj,  MR=y,  AD=H,  AB=2L,  weight 

of  cubic  foot  of  material  of  apca==^,,     Draw  ~Rm  a  perpen- 

dicular upon  PA  produced;  then  by  the  principle  of  the 

equality  of  moments, 

Em  .  P=MQ  .  (weight  of  DM). 


THE   PLATE   BANDE.  403 

But  Rm  =  x  cos.  a  —  y  sin.  a,  MQ  =  -Ja?,  weight  of  DM  = 
Hf^ ;  also  resolving  P  vertically, 

PcoB.*=LHf*I (405). 

Whence  we  obtain,  by  substitution  in  the  preceding  equa- 
tion, and  reduction, 

L(x— y  tan.  a) =•££' (406), 

which  is  the  equation  to  the  line  of  resistance,  showing  it  to 
be  a  parabola.  If,  in  this  equation,  L  be  substituted  for  a?, 
and  the  corresponding  value  of  y  be  represented  by  Y,  there 
will  be  obtained  the  equation  Y  tan.  a  =  |L,  whence  it 
appears  that  a  is  less  as  Y  is  greater ;  but  by  equation  (405), 
P  is  less  as  a  is  less.  P,  therefore,  is  less  as  Y  is  greater ; 
but  Y  can  never  exceed  H,  since  the  line  of  resistance  can- 
not intersect  the  extrados.  The  least  value  of  P,  consistent 
with  the  stability  of  the  plate  bande,  is  therefore  that  by 
which  Y  is  made  equal  to  H,  and  the  line  of  resistance 
made  to  touch  the  upper  surface  of  the  plate  bande  in  F. 

Now  this  least  value  of  P  is,  by  the  principle  of  least 
•resistance  (see  Theory  of  the  Arch\  the  actual  value  of  the 
resistance  at  A, 

/.tan.a^ijj (40Y). 

Eliminating  a  between  equations  (405)  and  (407), 

(408). 

Multiplying  equations  (405)  and  (407)  together, 
P  sin.  a^-JLX (409). 

Now  P  sin.  a  represents  the  horizontal  thrust  on  the  point 
of  support  A.  From  this  equation  it  appears,  therefore,  that 
the  horizontal  thrust  upon  the  abutments  of  a  straight  arch 
is  wholly  independent  of  the  depth  H  of  the  arch,  and  that 
it  varies  as  the  square  of  the  length  L  of  the  arch ;  so  that 
the  stability  of  the  abutments  of  such  an  arch  is  not  at  all 
diminished,  but,  on  the  contrary,  increased,  by  increasing 
the  depth  of  the  arch.  This  increase  of  the  stability  of  the 
abutment  being  the  necessary  result  of  an  increase  of  the 
vertical  pressure  on  the  points  of  support,  accompanied  by 
no  increase  of  the  horizontal  thrust  upon  them. 


404 


THE   PLATE   BANDE. 


308.  The  loaded  plate  lande. 

It  is  evident  that  the  effect  of  a  loading,  distributed 
uniformly  over  the  extrados  of  the 
plate  bande,  upon  its  stability,  is  in 
every  respect  the  same  as  would  be 
produced  if  the  load  were  removed, 
and  the  weight  of  the  material  of 
the  bande  increased  so  as  to  leave 
the  entire  weight  of  the  structure 
unchanged.  Let  ^3  represent  the 
weight  of  each  cubic  foot  when  thus 
increased,  M-2  the  weight  of  each 
cubic  foot  of  the  load,  and  Hx  the  height  of  the  load ;  then 


H 


(410). 


The  conditions  of  the  stability  of  the  loaded  plate  bande 
are  determined  by  the  substitution  of  this  value  of  ^  for  ^1 
in  the  preceding  article. 


309.  Conditions  necessary  that  the  voussoirs  of  a  plate  lande 
may  not  slip  upon  one  another. 

It  is  evident  that  the  inclination  of  every  other  resultant 
pressure  to  the  perpendicular  to  the  surface  of  its  corres- 
ponding joint,  is  less  than  the  inclination  of  the  resultant 

pressure  or  resistance  P,  to  the 
perpendicular  to  the  joint  AD. 
If,  therefore,  the  inclination  be 
not  greater  than  this  limiting  an- 
gle of  resistance,  then  will  every 
other  corresponding  inclination 
be  less  than  it,  and  no  voussoir 
will  therefore  slip  upon  the  sur- 
face of  its  adjacent  voussoir.  Now  the  tangent  of  the  incli- 
nation P  to  the  perpendicular  to  AD  is  represented  by  cot.  a 

2H 

or  by  -j-  (equation  40 7) ;  the  required  condition  is  therefore 

determined  by  the  inequality, 

?5<tan.9 (411). 


THE    SLOPING    BUTTRESS.  405 

It  is  evident  that  the  liability  of  the  arch  to  failure  by  the 
slipping  of  its  voussoirs,  is  less  as  its  depth  is  less  as  com- 
pared to  its  length.  In  order  the  more  effectually  to  pro- 
tect the  arch  against  it,  the  voussoirs  are  sometimes  cut  of 
the  forms  shown  by  the  dotted  lines  in  the  preceding  figure, 
their  joints  converging  to  a  point.  The  pressures  upon  the 
points  A  and  B  are  dependent  upon  the  form  of  that  portion 
of  the  arch  which  lies  between  those  points,  and  indepen- 
dent of  the  forms  of  the  voussoirs  which  compose  it  ;  these 
pressures,  and  the  conditions  of  the  equilibrium  of  the  piers 
which  support  the  arch,  remain  therefore  unchanged  by  this 
change  in  the  forms  of  the  voussoirs. 

310.  To  determine  the  conditions  of  the  equilibrium  of 
the  upright  piers  or  columns  of  masonry  which  form  the 
abutments  of  a  straight  arch,  supposing  them  to  be  termi- 
nated, as  shown  in  the  figure,  on  a  different  level  from  the 
extrados  CD  of  the  arch,  let  b  be  taken  t*o  represent  the 
elevation  of  the  top  of  the  pier  above  the  point  A  ;  then  will 

Z>  tan.  a,  or  |^j  (equation  407),  represent  the  distance  AG 

(p.  383),  or  the  value  of  Ic—  \a).  Substituting  for  k  in  equa- 
tion (377)  and  also  the  values  of  P  sin.  a,  P  cos.  a,  from 
equations  (409)  and  (405),  we  have 


(412); 


which  is  the  equation  to  the  line  of  resistance  of  the  pier,  a 
representing  its  thickness,  5  the  height  of  its  summit  above 
the  springing  A  of  the  arch,  L  the  length  of  the  arch,  f*  the 
weight  of  a  cubic  foot  of  the  material  of  the  arch  or  abut- 
ment (supposed  the  same). 

The  conditions  of  the  stability  may  be  determined  from 
this  equation  as  in  the  preceding  articles.  If  the  arch  be 
uniformly  loaded,  the  value  of  ^3  given  by  equation  (410) 
must  be  substituted  for  f. 


311.    THE    CENTRE    OF    GRAVITY    OF    A    BUTTRESS    WHOSE    FACES 
ARE   INCLINED   AT   ANY    ANGLE   TO    THE    VERTICAL. 

Let  the  width  AB  of  the  buttress  at  its  summit  be  repre- 


406 


THE    SLOPING   BUTTRESS. 


sented  by  #,  its  width  CD  at  the  base  by  lj 
its  vertical  height  AF  by  <?,  the  inclination 
of  its  outer  face  or  extrados  BC  to  the 
vertical  by  al9  that  of  its  intrados  AD  by 
aa. 

Let  II  represent  the  centre  of  gravity  of 
the  parallelogram  ADEB,  and  K  that  of 
the  triangle  BCE,  and  G  that  of  the  but- 
tress ;  draw  HM,  GL,  ICN",  perpendiculars 
upon  AF.  Then  representing  GL  by  \ 
and  observing  that  the  area  ADEB  is  represented  by 
the  area  EBC  by  %(b— a}c,  and  the  area  ADCB  by  £f 


Now  HM= 


n.  a2)  ; 
Substituting  these  values  and  reducing, 


a+b 

tan.  a^^a  +  c  tan.  aa), 
tan.  a,= 


J=CD=CF—  DF=c  tan.  a^a—  c  tan.  a2  ;  also  (a3  +  ab  +  62) 

—(b—a)*  +  Sab  =  <?2(tan.a,  —  tan.a2)2  +  3ac(tan.a1  —  tan.a2)  +  3^2, 

(a  +  25)<?tan.  aa=  |2c(tan.  a,1—  tan.a2)  +  3«(  c  tan.  aa 


c  tan.  aa 
=2c2  (tan.  OLI—  tan.  a2)  tan.  a2  +  3ac  tan.  a2  ; 

+  2£)  e  tan.  a2=c2  (tan.*^—  tan.  3a2) 
+  3ac  tan.  aj  +  3^2. 
'  8ai~tan.  2a2)  +  ac  tan.  ax 


a!—  tan.  a2) 


.  .  .  (414). 


312.  THE  LINE  OF  RESISTANCE  IN  A  BUTTRESS. 

Let  LM  represent  any  horizontal  section  of  the  buttress, 
TK  a  vertical  line  through  the  centre  of 
£ .Ji  gravity  of  that  portion  AMLB  of  the  but- 
tress which  rests  upon  this  section.  Pro- 
duce LM  to  meet  the  vertical  AE  in  V, 
and  let  KY=^  and  AV=a? ;  then  is  the 
value  of  X  determined  by  substituting  x 
for  c  in  equation  (414).  Let  PO  be  the 
direction  in  which  a  single  pressure  P  is 
applied  to  overturn  the  buttress.  Take 

*  This  equation  is,  of  course,  to  be  adapted  to  the  case  in  which  the  inclina- 
tion of  AD  is  on  the  other  side  of  the  vertical,  as  shown  by  the  dotted  line 
Ac?  by  making  «2,  and  therefore  tan.  a2  negative. 


THE    SLOPING   BUTTRESS.  407 

OS  to  represent  P  in  magnitude  and  direction,  and  ON  to 
represent  the  weight  of  the  portion  AMLB  of  the  buttress  ; 
complete  the  parallelogram  SN,  and  produce  its  diagonal 
OR  to  Q ;  then  will  OR  evidently  be  the  direction  01  the 
resultant  pressure  upon  AMLB5  and  Q  a  point  in  the  line  of 
resistance. 

Let  YQr=y,  AG=&,  /GOT— i,  (*:=  weight  of  each  cubic 
foot  of  material ;  and  let  the  same  notation  be  adopted  in 
other  respects  as  in  the  last  article.  By  similar  triangles, 

QK_RI 
OK"  01 


OK=TK— TO=TK— TG  cot.  GOT=ra;— (X+&)  cot.  i,. 
HI— EN  sin.  ROT=P  sin.  i, 

OI=OISr+OT=iaAV(AB+LM)-f  RN  cos.  RNI= 
Jf*aj{2a4-iB  (tan.  c^ — tan.  a2)J  +P  cos.  £ ; 

y— X P  sin,  t 

"x — (X  +  #)  cot.  i    ^px{%a-}-x(t&n.  ax — tan.  aa)}-f-P  cos.  i 

Transposing  and  reducing, 

_jXfxa?  \%a+x  (tan,  otj— tan.  a2)^  +P  (a?  sin,  i— 7c  cos.  1 1 
•J^{2«-fa?(tan.  «,— tan.  a.2)\  +P  cos.  t        ~' 

but  substituting  x  for  <?  in  equation  (414),  and  multiplying 
both  sides  of  that  equation  by  the  denominator  of  the  frac- 
tion in  the  second  member,  and  by  the  factor  4a#5  we  have 


(tan.  a,—  tan.  a2)}  =^x3  (tan.Vj—  tan.  2 
tan. 


n.^!—  tan.'q^-j-^atan.  «x-f-//a;a24-2P(a;  sin,  f—  ^  cos,  i) 
Hx  I2a4-actan.d-tan.aa)!  -f-2P  cos.  i 

which  is  the  equation  to  the  line  of  resistance  in  a  buttress. 
If  the  in  trades  AD  be  vertical,  tan.  a2  is  to  be  assumed  =0. 
If  AD  be  inclined  on  the  opposite  side  of  the  vertical  to  that 
shown  in  the  figure,  tan.  «2  is  to  be  taken  negatively.  The 
line  of  resistance  being  of  three  dimensions  in  x,  it  follows 
that,  for  certain  values  of  y,  there  are  three  possible  values 
of  x  ;  the  curve  has  therefore  a  point  of  contrary  flexure. 
The  conditions  of  the  equilibrium  of  the  buttress  are  deter- 


408  WALL    SUSTAINING   THE 

mined  from  its  Hne  of  resistance  precisely  as  those  of  the 
wall. 

Thus  the  thickness  a  of  the  buttress  at  its  summit  being 
given,  and  its  height  c,  and  it  being  observed  that  the  dis- 
tance CE  is  represented  by  a+c  tan.  a1?  the  inclination  o^  of 
its  extrados  to  the  vertical  may  be  determined,  so  that  its 
line  of  resistance  may  intersect  its  foundation  at  a  given  dis- 
tance m  from  its  extrados,  by  solving  equation  (415)  in  re- 
spect to  tan.  «1?  having  first  substituted  c  for  x  and  a  +  c  tan. 
^—m  for  y\  and  any  other  of  the  elements  determining  the 
conditions  of  the  stability  of  the  buttress  may  in  like  manner 
be  determined  by  solving  the  equation  (the  same  substitu- 
tions being  made  in  it)  in  respect  to  that  element. 


313.     A   WALL   OF   UNIFORM    THICKNESS     SUSTAINING    THE   FEES- 
SURE   OF   A   FLUID. 

If  E  be  taken  to  represent  the  surface  of  the  fluid,  IK  any 
section  of  the  wall,  and  EP  two  thirds 
Isss^—  ~=r=  t^le  dePtn  EK  ;  then  will  P  be  the  cen- 
tre of  pressure*  of  EK,  the  tendency 
of  the  fluid  to  overturn  the  portion 
AKIB  of  the  wall  being  the  same  as 
would  be  produced  by  a  single  pressure 
applied  perpendicular  to  its  surface  at 
P,  and  being  equal  in  amount  to  the 
weight  of  a  mass  of  water  whose  base 
is  equal  to  EK,  and  its  height  to  the 
depth  of  the  centre  of  gravity  of  EK,  or 

to  ^EK.     Let  AK=#,  AE=e,  weight  of  each  cubic  foot  of 

the  fluid—  f* 


Let  the  direction  of  P  intersect  the  axis  of  the  wall  in  O  ; 
let  it  be  represented  in  magnitude  by  OS  ;  take  ON  to 
represent  the  weight  of  the  portion  AKIB  of  the  wall  ;  com- 
plete the  parallelogram  SIN",  and  produce  its  diagonal  to 
meet  IK  in  Q  ;  then  will  Q  be  a  point  in  the  line  of  resist- 
ance. Let  QM=y,  AB=#,  weight  of  each  cubic  foot  of 

material  of  wall=fA.     By  similar  triangles,   r    =      :.     Now 


*  Treatise  on  "  Hydrostatics  and  Hydrodynamics,"  by  the  author  of  this 
work,  Art.  38.  p.  26. 


PRESSURE   OF   A   FLUID.  4()9 


—e)',  NO  =  weight 


Dividing  numerator  and  denominator  of  this  equation  by 

M, 

jxl5  and  observing  that  the  fraction  —  represents  the  ratio  o 

*i 

of  the  specific  gravities  of  the  material  of  the  wall  and  the 
fluid,  we  have 


which  is  the  equation  to  the  line  of  resistance  in  a  wall  of 
uniform  thickness,  sustaining  the  pressure  of  a  fluid. 


314.  To  determine  the  thickness,  a,  of  the  wall,  so  that  its 
height,  h,  being  given,  the  line  of  resistance  may  intersect 
its  foundation  at  a  given  distance,  m,  within  the  extrados. 

Substituting,  in  equation  (416),  h  for  a?,  and  \a— m  for  y, 
and  solving  the  resulting  equation  in  respect  to  a,  we  obtain 

|/~ i  ^~6)3 

Equation    (416)    may    be    put    under    the    form    y= 

1       /         e\ 3 

~ — a?2 1 1  —  -1  ;  whence  it  is  apparent  that  y  increases  con- 
tinually with  x ;  so  that  the  nearest  approach  is  made  by 
the  line  of  resistance,  to  the  extrados  of  the  pier,  at  its 
lowest  section,  m  therefore  represents,  in  the  above  expres- 
sion, the  modulus  of  stability  (Art.  286). 


315.  The  conditions  necessary  that  the  wall  should  not  be 
overthrown  by  the  slipping  of  the  courses  of  stones  on  one 
another. 

The  angle  SEO  represents  the  inclination  of  the  resultant 
pressure  upon  the  section  IK  to  the  perpendicular ;  the  pro- 
posed condition  is  therefore  satisfied,  so  long  as  SRO  is  less 
than  the  limiting  angle  of  resistance  9. 


410 


Now,  tan. 


WALL   SUSTAINING   THE 

OS 


;  the  proposed  con* 


(x — tf)2 
dition  is  therefore  satisfied,  so  long  as     ^  —  <  tan.  <p ;   or, 

reducing  this  inequality,  so  long  as 


316.   THE   STABILITY   OF   A   WALL    OF  (VARIABLE    THICKNESS 

SUSTAINING   THE   PRESSURE   OF   A   FLUID. 

Let  us  firsteuppose  the  internal  face  AB  of  the  wall  to  be 
vertical  ;  let  XY  be  any  section  of  it, 
P  the  centre  of  pressure  of  EX,  and 
SM  a  vertical  through  the  centre  of 
gravity  of  the  portion  AXYD  of  the 
wall.  Produce  the  horizontal  direc- 
tion of  the  pressure  P  of  the  fluid, 
supposed  to  be  collected  in  its  centre 
of  pressure,  to  meet  MS  in  S,  and  let 
SK  be  taken  to  represent  it  in  mag- 
nitude, and  ST  to  represent  the  weight 
of  the  portion  AXYD  of  the  wall, 
and  complete  the  parallelogram  STRK  ;  then  will  its 
diagonal  SR  represent  the  direction  and  amount  of  the 
resultant  pressure  upon  the  mass  AXYD,  and  if  it  be  pro- 
duced to  intersect  XY  in  Q,  Q  will  be  a  point  in  the  line  of 
resistance. 

Let  AX=#,  XQ=2/,  MX=X,  AE=0,  AD=a,  inclination 
of  DC  to  vertical—  a,  f*=  weight  of  cubical  foot  of  wall, 
fjil=  weight  of  cubical  foot  of  fluid.  By  similar  triangles, 
QM_RT 

~*     '     W 


RT=pressure  of  fluid  on  EX=iEX.f*1EX=i*I(»— 
ST=weight  of  mass  AY=-|-{2#+a?  tan.  a\x\*>. 


*  The  centre  of  pressure  of  a  rectangular  plane  surface  sustaining  the 
pressure  of  a  fluid  is  situated  at  two  thirds  the  depth  of  its  immersion.  — 
Hydrostatics,  p.  26. 

f  The  pressure  of  a  heavy  fluid  on  any  plane  surface  is  equal  to  the  weight 
of  a  prism  of  the  fluid  whose  base  is  equal  in  area  to  the  surface  pressed,  and 
its  height  to  the  depth  of  the  centre  of  gravity  of  the  surface  pressed.  — 
Hydrostatics,  Art.  31. 


PRESSURE   OF   A  FLUID.  411 


M, 

Let—  =0;  then,  if  the  fluid  be  water,  a  represents  the 

specific  gravity  of  the  material  of  the  wall  ;  and  if  not,  it 
represents  the  ratio  of  the  specific  gravities  of  the  fluid  and 
wall. 

(x-ef 


._ 

- 


tan.  a* 


K"ow  making  aa=0  in  equation  (414),  and  substituting  a 
for  aa  and  x  for  <?, 

^a?2  tan.  2a  4-  ax  tan.  a  +  a*  _-J#>8  tan.  2a  +  aw*  tan.  a  +  #2# 

x  tan.  a  +  2#  %OjX  +  a?2  tan.  a 

Adding  this  equation  to  the  preceding, 

o-(aj—  •  e)3-}-^xs  tan.V  +  ^a?2  tan.  a+ofte 


which  is  the  equation  to  the  line  of  resistance  to  the  wall, 
the  conditions  of  whose  stability  may  be  determined  from  it 
as  before  (see  Arts.  291.  293.). 


31T.  The  conditions  necessary  iJiat  no  course  of  stones  com- 
posing the  wall  may  slip  upon  the  subjacent  course. 

This  condition  is  satisfied  when  the  inclination  of  SQ  to 
the  perpendicular  to  the  surface  of  contact  at  Q  is  less  than 
the  limiting  angle  of  resistance  9 ;  that  is,  when  QSM  <<p, 
or  when 

ET 
tan.  9>tan.  QSM,  or  >,  or  > 


or  tan.  9  >  I- 

ISTo  course  of  stones  will  be  made  by  the  pressure  of  the 
fluid  to  slip  upon  the  subjacent  course  so  long  as  this  condi- 
tion is  satisfied. 

It  is  easily  shown  that  the  expression  forming  the  second 
member  of  the  above  inequality  increases  continually  with 


4:12  THE  NATURAL   SLOPE   OF  EARTH. 

a?,  so  that  the  obliquity  of  the  resultant  pressure  upon  each 
course,  and  the  probability  of  its  being  made  to  slip  upon 
the  next  subjacent  course,  is  greater  in  respect  to  the  lower 
than  the  upper  courses,  increasing  with  the  depth  of  each 
course  beneath  the  surface  of  the  fluid. 


EARTH  WORKS. 
318.  The  natural  slope  of  earth. 

It  has  been  explained  (Art.  241.)  that  a  mass,  placed  upon 
an  inclined  plane  and  acted  upon  by  no  other  forces  than  its 
weight  and  the  resistance  of  the  plane,  will  just  be  supported 
when  the  inclination  of  the  plane  to  the  horizon  equals  the 
limiting  angle  of  resistance  between  the  surface  of  the  plane 
and  that  of  the  mass  which  it  supports ;  so  that  the  limiting 
angle  of  resistance  between  the  surfaces  of  the  component 
parts  of  any  mass  of  earth  might  be  determined  by  varying 
continually  the  slope  of  its  surface  until  a  slope  or  inclination 
was  attained,  at  which  particular  slope  small  masses  of  the 
same  earth  would  only  just  be  supported  on  its  surface,  or 
would  just  be  upon  the  point  of  slipping  down  it.  Now  this 
process  of  experiment  is  very  exactly  imitated  in  the  case  of 
embankments,  cuttings,  and  other  earth-works,  by  natural 
causes.  If  a  slope  of  earth  be  artificially  constructed  at  an 
inclination  greater  than  the  particular  inclination  here 
spoken  of,  although,  at  first,  the  cohesion  of  the  material 
may  so  bind  its  parts  together  as  to  prevent  them  from  slid- 
ing upon  one  another,  and  its  surface  from  assuming  its 
natural  slope,  yet  by  the  operation  of  moisture,  penetrating 
its  mass  and  afterwards  drying,  or  under  the  influence  of 
frost,  congealing,  and  in  the  act  of  congelation  expanding 
itself,  this  cohesion  of  the  particles  of  the  mass  is  continually 
in  the  process  of  being  destroyed ;  and  thus  the  particles,  so 
long  as  the  slope  exceeds  the  limiting  angle  of  resistance, 
are  continually  in  the  act  of  sliding  down,  until,  when  that 
angle  is  at  length  reached,  this  descent  ceases  (except  in  so 
far  as  the  particles  continue  to  be  washed  down  by  the  rain), 
and  the  surface  retains  permanently  its  natural  slope. 

The  limiting  angle  of  resistance  9  is  thus'detennined  by 
observing  what  is  the  natural  slope  of  each  description  of 
earth. 


THE  PRESSURE   OF  EARTH. 


4:13 


The  following  table  contains  the  results  of  some  such 
observations  * : — 


NATURAL  SLOPES  OF  DIFFERENT  KINDS  OF  EARTH. 


Nature  of  Earth. 

Natural  Slope. 

Authority. 

Fine  dry  sand  (a  single  experiment)  • 
Ditto  
Ditto  
Common  earth  pulverised  and  dry    - 
Common  earth  slightly  damp     - 
Earth  the  most  dense  and  compact   - 
Loose  shingle  perfectly  dry 

21° 
34°  29' 
39° 
46°  50' 
54° 
55° 
39° 

Gadroy. 
Ron  dele  t. 
Barlow. 
Rondelet. 
Rondelet. 
Barlow. 
Pasley. 

SPECIFIC  GRAVITIES  OF  DIFFERENT  KINDS  OF  EARTH. 


Nature  of  Earth. 

Specific  Gravity. 

I  ,A 

1  .(• 

Marl          

1-9 
1*7 

Rubble  masonry  of  calcareous  earth  or  siliceous  stones 
Rubble  masonry  of  granite        ..... 
Rubble  masonry  of  basaltic  stones    .... 

1-7  to  2-3 
2-3 
2-5 

319.  THE  PRESSURE  OF  EARTH. 

Let  BD  represent  the  surface  of  a  wall  sustaining  the 
pressure  of  a  mass  of  earth  whose  surface  AE  is  horizontal. 

Let  P  represent  the  resultant  of  the  pressures  sustained 
by  any  portion  AX  of  the  wall ;  a*nd  let  the  cohesion  of  the 
particles  of  the  earth  to  one  another  be  neglected,  as  also 
their  friction  on  the  surface  of  the  wall.  It  is  evident  that 


*  It  is  taken  from  the  treatise  of  M.  Navier,  entitled  Resume  (Fun  Cours  ck 
Construction,  p.  160.  \ 


414 


THE   PRESSURE   OF   EARTH. 


any  results  deduced  in  respect  to 
the  dimensions  of  the  wall,  these 
._„.*  elements  of  the  calculation  being 
.*;  neglected,  will  be  in  excess^  and 

err  on  the  safe  side. 
?  Now  the  mass  of  earth  which 
presses  upon  AX  may  yield  in  the 
direction  of  any  oblique  section 
XY,  made  from  X  to  the  surface 
AE  of  the  mass.  Suppose  YX  to 
be  the  particular  direction  in  which 
it  actually  tends  to  yield  ;  so  that 
if  AX  were  removed,  rupture 
would  first  take  place  along  this 
section,  and  AXY  be  the  portion  of  the  mass  which  would 
first  fall.  Then  is  the  weight  of  the  mass  AYX  supported 
by  the  resistances  of  the  different  elements  of  the  surface  AX 
of  the  wall,  whose  resultant  is  P,  and  by  the  resistance  of 
the  surface  XY  on  which  it  tends  to  slide.  Suppose,  now, 
that  the  mass  is  upon  the  point  of  sliding  down  the  plane 
XY,  the  pressure  P  being  that  only  which  is  just  sufficient 
to  support  it ;  the  resultant  SE  of  the  resistances  of  the 
different  points  of  XY  is  therefore  inclined  (Art.  241.)  to  the 
normal  ST,  at  an  angle  RST  equal  to  the  limiting  angle  of 
resistance  9  between  any  two  contiguous  surfaces  of  the 
earth. 

Now  the  pressure  P,  the  weight  "W  of  the  mass  AXY,  and 
the  resistance  E,  being  pressures  in  equilibrium,  any  two  of 
them  are  to  one  another  inversely  as  the  sines  of  their  incli- 
nations to  the  third  (Art.  14.). 


**w 


sin.  WSE 
:sin.  PSE 


sin.  WSE 
sin.  PSE  ' 


But  WSE=WST-EST=AYX-EST=^-<-<pT 


=<;  PSE=PST+EST==AXY+KST=*+<p. 

....  (42.1). 


if 


Also  W=-J^AX  .  AY^J-^a?3  tan.  i  ;  if  ^l=  weight   of  each 
cubic  foot  of  earth,  and  AX=a?  ; 


.•.P^z-Jfjo^8  tan.  i  cot.  (*  +  <?)  .  .  .  .  (422). 
Now  it  is  evident  that  this  expression,  which  represents 


THE   PRESSURE   OF   EARTH.  4:15 

the  resistance  of  the  wall  necessary  to  sustain  the  pressure  of 
the  wedge-shaped  mass  of  earth  AXY,  being  dependent  for 
its  amount  upon  the  value  of  *  (so  that  different  sections, 
such  as  XY,  being  taken,  each  different  mass  cut  off  by  such 
section  will  require  a  different  resistance  of  the  wall  to  sup- 
port it),  may  admit  of  a  maximum  value  in  respect  to  that 
variable.  *  And  if  the  wall  be  made  strong  enough  to  supply 
a  resistance  sufficient  to  support  that  wedge-shaped  mass  of 
earth  whose  inclination  i  corresponds  to  the  maximum  value 
of  P,  and  which  thus  requires  the  greatest  resistance  to  sup- 
port it  ;  then  will  the  earth  evidently  be  prevented  by  it  from 
slipping  at  any  inclination  whatever,  for  it  will  evidently  not 
slip  at  that  angle,  the  resistance  necessary  to  support  it  at 
that  angle  being  supplied  ;  and  it  will  not  slip  at  any  other 
angle,  because  more  than  the  resistance  necessary  to  prevent 
it  slipping  at  any  other  angle  is  supplied. 

If,  then,  the  wall  supplies  a  resistance  equal  to  the  maxi- 
mum value  of  P  in  respect  to  the  variable  i>,  it  will  not  be 
overthrown  by  the  pressure  of  the  earth  on  AX.  Moreover, 
if  it  supply  any  less  resistance,  it  will  be  overthrown  ;  there 
not  being  a  sufficient  resistance  supplied  by  it  to  prevent  the 
earth  from  slipping  at  that  inclination  i  which  corresponds 
to  the  maximum  value  of  P. 

To  determine  the  actual  pressure  of  the  earth  on  AX,  we 
have  then  only  to  determine  the  maximum  value  of  P  in  re- 
spect to  i. 

This  maximum  value  is  that  which  satisfies  the  conditions 

dP 


But  differentiating  equation  (422)  in  respect  to  t,  we  obtain 
by  reduction 


,  . 

€U  cos.  i  sin.  (1+9) 

Let  the  numerator  and  denominator  of  the  fraction  in  the 

*  The  existence  of  this  maximum  will  subsequently  be  shown  :  it  is,  how- 
ever, sufficiently  evident,  that,  as  the  angle  i  is  greater,  the  wedge-shaped  mass 
to  be  supported  is  heavier;  for  which  cause,  if  it  operated  alone,  P  would  be- 
come greater  as  L  increased.  But  as  i  increases,  the  plane  XY  becomes  less 
inclined;  for  which  cause,  if  it  operated  alone,  P  would  become  less  as  L  in 
creased.  These  two  causes  thus  operating  to  counteract  one  another,  deter- 
mine a  certain  inclination  in  respect  to  which  their  neutralising  influence  is  the 
least,  and  P  therefore  the  greatest. 

f  Church's  Diff.  and  Int.  Cal.,  Art.  41. 


416  REVETEMENTS. 

second  member  of  this  equation  be  represented  respectively 
by  p  and  q  ;  therefore  -^r=i^1ar'  .  -,  i^-q  ~~-/p\  ;  but  when 

—=-  =0,  p=0  ;  in  this  case,  therefore,  -=-5-=  J(*  a?8--^  .  Whence 
w>  cLi  di 


it  follows,  by  substitution,  that  for  every  value  of  i  by  which 
the  first  condition  of  a  maximum  is  satisfied,  the  second  dif- 
ferential co-efficient  becomes 


Now  it  is  evident  from  equation  (423)  that  the  condition 
-y-=0  is  satisfied  by  that  value  of  t  which  makes  2(*-j-(p)=: 

(il 

*— 2«,  or 

<=H w 

And  if  this  value  be  substituted  for  i  in  equation  (424),  it 
becomes 


J«    9\  •     ./*    <P 

cos-  (4-2)  ^  b+i 


which  expression  is  essentially  negative,  so  that  the  second 
condition  is  also  satisfied  by  this  value  of  L.  It  is  that,  there- 
fore, which  corresponds  to  the  maximum  value  of  P  ;  and 
substituting  in  equation  (422),  and  reducing,  we  obtain  for 
this  maximum  value  of  P  the  expression 


which  expression  represents  the  actual  pressure  of  the  earth 
on  a  surface  AX  of  the  wall,  whose  width  is  one  foot  and  its 
depth  x. 


REVETEMENT  WALLS. 
320.    If,  instead  of  a  revetement  wall  sustaining  the  pres- 


KEVETEMENTS. 


417 


sure  of  a  mass  of  earth,  the  weight 
Y       K    of  each  cubic  foot  of  which  is  re- 
tt:.r?t?r^wp:$jg:*jl    presented  by  f^,  it  had  sustained 
$1    the  pressure  of  a  fluid,  the  weight 
/  |)    of  each  cubic  foot  of  which  was  re. 

$*/  F    presented  by  f*r  tan. 2 1  -— - 1 ,  then 

\  !  /  \4r       2if 

"fi?  |i     would  the  pressure  of  that  fluid 

'  |  V  $     upon  the  surface  AX  have  been 

S!     represented*  by  J|mXtan-a  ( -— -1 

>**.  \  4:       4'A 

.•••";?=g?.»; ft»»»v,ugjs^  go  ijj^  ^g  pressure  of  a  mass  o^ 
earth  upon  a  revetement  wall  (equation  427),  when  its,  sur- 
face is  horizontal  (and  when  its  horizontal  surface  extends,, 
as  shown  in  the  figure,  to  the  very  surface'  of  the  wall)^is 
identical  with  that  of  an  imaginary  fluid  whose  specific  gra 
vity  is  such  as  to  cause  each  culic  foot  of  it  to  have  a  weight 
M-j,  represented  in  pounds  by  the  formula 


Substituting  this  value  far  ^  in  equations  (416)  and  (419), 
we  determine  therefore,  at  once,  the  lines  of  resistance  in 
revetement  walls  of  uniform  and  variable  thickness,  under 
the  conditions  supposed,  to  be  respectively 


—tan. a  I  -_ | J(x— e)a -h-^'tan.V  +  aa?atan.a -f  cfx 
11-=.  — — — — 

&  O/v/vj     I     /vj2   4-^.-n 


..(430); 


where  a  represents  the  ratio  of  the  specific  gravity  of  the 
material  of  the  wall  to  that  of  the  earth.  The  conditions  of 
the  equilibrium  of  the  revetement  wall  may  be  determined 
from  the  equation  to  its  line  of  resistance,  as  explained  in 
the  case  of  the  ordinary  wall. 


27 


Hydrostatics,  Art.  31. 


418 


REVETEMENTS. 


321.  The  conditions  necessary  that  a  revetement  wall  may 
not  le  overthrown  ~by  the  slipping  of  the  stones  of  any 
course  upon  those  of  the  subjacent  course. 

These  are  evidently  determined  from  the  inequality  (420) 
by  substituting  /*„  (equation  428)  for  ^  in  that  inequality  ; 
we  thus  obtain,  representing  the  limiting  angle  of  resistance 
of  the  stones  composing  the  wall  by  <PI  to  distinguish  it  from 
that  9  of  the  earth, 


where  a  represents  the  ratio  of  the  specific  gravity  of  the 
material  of  the  wall  to  that  of  the  earth. 

As  before,  it  may  be  shown  from  this  expression  that  the 
tendency  of  the  courses  to  slip  upon  one  another  is  greater 
in  the  lower  courses  than  the  higher. 


322.  The  pressure  of  earth  whose  surface  is  inclined  to  the 

horizon. 

Let  AB  represent  the  surface  of  such  a  mass  of  earth,  YX 

the  plane  along  which  the 
rupture  of  the  mass  in 
contact  with  the  surface 
AX  of  a  revetement  wall 
tends  to  take  place,  AX= 
a,  AXY=*,  XAB=/3. 
Then  if  W  be  taken  to 
represent  the  weight  of 
the  mass  AXY,  it  may  be 
shown,  as  in  Art,  319, 
equation  (421),  that  P= 
W  cot, 


„         tTT 
fore    W= 


=frl  AX.  AY.  sin.   0, 
#a  sin.  i  sin.  /3 


;  there- 


cot. 


Now  the  value  of  i  in  this  function  is  that  which  renders 
it  a  maximum  (Art,  31^).     Expanding  cot,  (1  +  9),  and  dif- 


REVETEMENTS. 


419 


ferentiating  in  respect  to  tan.  t,  this  value  of  i  is  readily 
determined  to  be  that  which  satisfies  the  equation 

cot.  t=tan.  (p  +  sec.  9  4/r+cot7/3  cot.  9  ....  (433). 
Substituting  in  equation  (432),  and  reducing, 

{COS    (p  I 

l  +  sin.^l  +  cot.gcot.M  .....  (43*> 

From  which  equation  it  is  apparent,  that  the  pressure  of  the 
earth  is,  in  this  case,  identical  with  that  of  a  fluid,  of  such  a 
density  that  the  weight  j*a,  of  each  cubic  foot  of  it,  is  repre- 
sented by  the  formula 


1  +  sin.  9  v  1+cot.  <f>  cot.  /3 


(435). 


The  conditions  of  the  equilibrium  of  a  revetement  wall 
sustaining  the  pressure  of  such  a  mass  of  earth  are  therefore 
determined  by  the  same  conditions  as  those  of  the  river  wall 
(Arts.  313  and  316). 


323.  THE  RESISTANCE  or  EAKTH. 

Let  the  wall  BDEF  be  supported  by  the  resistance  of  a 

mass  of  earth  upon  its  sur- 
face AD,  a  pressure  P,  ap- 
plied to  its  opposite  face, 
tending  to  overthrow  it.  Let 
the  surface  AH  of  the  earth 
be  horizontal ;  and  let  Q 
represent  the  pressure  which, 
being  applied  to  AX,  would 
just  be  sufficient  to  cause  the 
mass  of  earth  in  contact 
with  that  portion  of  the  wall 
to  yield ;  the  prism  AXY 
slipping  backwards  upon  the 
surface  XY.  Adopting  the  same  notation  as  in  Art.  319, 
and  proceeding  in  the  same  manner,  but  observing  that  US 
is  to  be  measured  here  on  the  opposite  side  of  TS  (Art.  241), 
since  the  mass  of  earth  is  supposed  to  be  upon  the  point  of 
slipping  upwards  instead  of  downwards,  we  shall  obtain 


^-l^aj8  tan.  i  cot.  (4—9) (436). 


420 


WALLS   BACKED   BY   EARTH. 


Now  it  is  evident  that  XY  is  that  plane  along  which  rup- 
ture may  be  made  to  take  place  by  the  least  value  of  Q  ;  / 
in  the  above  expression  is  therefore  that  angle  which  gives 
to  that  expression  its  minimum  value.  Hence,  observing 
that  equation  (436)  differs  from  equation  (422)  only  in  the 
sign  of  <p,  and  that  the  second  differential  (equation  426)  is 
rendered  essentially  positive  by  changing  the  sign  of  9,  it  is 
apparent  (equation  427)  that  the  value  of  Q  necessary  to 
overcome  the  pressure  of  the  earth  upon  AX  is  represented 


324.  It  is  evident  that  a  fluid  would  oppose  the  same 
resistance  to  the  overthrow  of  the  wall  as  the  resistance  of 
the  earth  does,  provided  that  the  weight  f*4  of  each  cubic 
foot  of  the  fluid  were  such  that 


(V=Man.'     +       ....(438); 

so  that  the  point  in  AX  at  which  the  pressure  Q  may  be 
conceived  to  be  applied,  is  situated  at  fds  the  distance  AX. 


325.  The  stability  of  a  wall  of  uniform  thickness  which  a 
given  pressure  P  tends  to  overthrow,  and  which  is  sus- 
tained by  the  resistance  of  earth. 

Let  y  be  the  point  in  whidi  any  section  XZ  of  the  wall 

would  be  intersected  by  the 
resultant  of  the  pressures 
upon  the  wall  above  that  sec- 
tion, if  the  whole  resistance 
Q,  which  the  earth  in  con- 
tact with  AX  is  capable  of 
supplying,  were  called  into 
action.  Let  BX=a?,  ~Ky=y, 
BA  =  e,  BE=0,  Bp  =  jfc, 
weight  of  cubic  feet  of  ma- 
terial of  wall— fA,  inclination 
of  P  to  vertical^.  Taking 
the  moments  about  the  point 
y  of  the  pressures  applied  to  BXZE,  we  have,  by  the  prin- 
ciple of  the  equality  of  moments,  observing  that  "'~  " 


WALLS  BACKED  BY  EARTH. 


421 


(a?— e),  and  that  the  perpendicular  from  y,  upon  P  is  repre- 
sented by  x  sin.  &—(k—y)  cos.  d, 


r  \x  sin.  d— (fc— y)  cos.  e\  =$(x—( 

or  substituting  for  Q  its  value  (equation  437),  and  solving  in 
respect  to  y, 

irM'4(aJ— e}*+^a?x— P(a?  sin.  6— Jc,  cos.  6) 

y±=. — =r (4o9). 

JL  cos.  o  -j-  \*<ax 

Now  it  is  evident  that  the  wall  will  not  be  overthrown 
upon  any  section  XZ,  so  long  as  the  greatest  resistance  Q, 
which  the  superincumbent  earth  is  capable  of  supplying,  is 
sufficient  to  cause  the  resultant  pressure  upon  EX  to  inter- 
sect that  section,  or  so  long  as  y  in  the  above  equation  has 
a  positive  value ;  moreover,  that  the  stability  of  the  wall  is 
determined  by  the  minimum  value  of  y  in  respect  to  x  in 
that  equation,  and  the  greatest  height  to  which  the  wall  can 
be  buut,  so  as  to  stand,  by  that  value  of  x  which  makes  y=0. 


326.  The  stability  of  a  wall  which  a  given  pressure  tends  to 
overthrow,  and  which  is  supported  lyy  a  mass  of  earth 
whose  surface  is  not  horizoni  " 


Let  (3  represent  the  inclination  of  the  surface  AB  of  earth 

to  the  horizon.  By  reasoning 
similar  to  that  of  Art.  322.,  it  is 
apparent  that  the  resistance  Q 
of  the  earth  in  contact  with  any 
given  portion  AX  of  the  wall  to 
displacement,  is  determined  by 
assigning  to  9  a  negative  value 
in  equation  (434).  Whence  it 
follows,  that  this  resistance  is 
equivalent  to  that  which  would 
be  produced  by  the  pressure  of 
a  fluid  upon  the  wall,  the  weight 
^6  of  each  cubic  foot  of  which 
was  represented  by  the  formula 


% 


A  i/ 

icos.  9  ) 2 
T  Y 
1— sin.  9  1/1  — cot.  9  cot.  j3  ) 


(440). 


The  conditions  of  the  stability  of  an  upright  wall  sub- 
jected to  any  given  pressure  P  tending  to  overthrow  it,  and 


422 


REVETEMENTS. 


sustained  by  the  pressure  of  such  a  mass  of  earth,  are  there- 
fore precisely  the  same  as  those  discussed  in  the  last  article  ; 
the  symbol  ^4  (equation  439)  being  replaced  by  |u-5  (equation 
440). 


327.  The  stability  of  a  revetement  wall  whose  interior  face 
is  inclined  to  the  vertical  at  any  angle  ;  taking  into  account 
the  friction  of  the  earth  upon  the  face  of  the  wall. 

Let  «2  represent  the  inclination  of  the  face  BD  of  such  a 
wall  to  the  vertical,  <p2  the  limiting  angle  of  resistance 
between  the  mass  of  earth  and  the  surface  of  the  wall ;  and 
let  the  same  notation  be  adopted  as  in  the  last  article  in 
respect  to  the  other  elements  of  the 
question,  and  the  same  construction 
made.  Draw  PQ  perpendicular  to  BD ; 
then  is  the  direction  PS  of  the  resist- 
ance of  the  wall  upon  the  mass  of  earth, 
evidently  inclined  to  QP  at  an  angle 
QPS  equal  to  the  limiting  angle  of 
resistance  <?2,  in  the  state  bordering 
upon  motion  by  the  overthrow  of  the 
wall*  (Art.  241.). 

Draw  Pn  horizontally  and  X$  verti- 
cally, produce  TS  and  ES  to  meet  it  in 
m  and  n,  and  let 

P 
W 


sin.  WSE     sin.  (WST-TSE) 
sin.  PSK  —  sin.  (EmP  +  SPm). 


But 


=  AYX=    - 


=  -  -t,  TSE=<p, 
2 


Also  W= 


sn.   t  +  9  +  9a  +  a2       sn.  ^ 

^^X  (tan.  i  +  tan.  aa)  ;  if  «X=a?, 


*  It  is  not  only  in  the  state  of  the  wall  bordering  upon  motion  that  thia 
direction  of  the  resistance  obtains,  but  in  every  state  in  which  the  stability  of 
the  wall  is  maintained.  (See  the  Principle  of  'Least  Resistance.) 


REVETEMENT8.  4:23 

aa)  ;  ; 

Assuming  aa-f-9+9a=/3,  then  differentiating  in  respect  to  t, 

7T> 

and  assuming  -^-  =  0,  we  -obtain  by  reduction 

—(tan.  i  +  tan.  aa)  cos.  (£—  9)  -f 

cos.  (t-f  (p)sin.  (*+/3)sec.  a*=0;  or, 

—  (tan.  i  +  tan.  aa)  (1  +  tan.  (3  tan.  (p)  -f- 

(1—  tan.  i  tan.  (p)  (tan.  i+  tan.  /3)=0  ; 

.'.  tan.3  1  +  2  tan.  i  tan.  /3  —  tan.  (3  cot.  9  + 

(cot.  9  +  tan.  /3)  tan.  aa  =  0. 

Solving  this  quadratic  in  respect  to  tan.  £,  neglecting  the 
negative  root,  since  tan.  i  is  essentially  positive,  and  reducing, 

tan.  i—  (tan.  /3—  tan.  a,)*(tan.  £  +  cot.  9)*—  tan.  /3  .  .  .  (442.) 

Now  the  value  of  t  determined  by  this  equation,  when 
substituted  in  the  second  differential  coefficient  of  P  in 
respect  to  £,  gives  to  that  coefficient  a  negative  value  ;  it 
therefore  corresponds  to  a  maximum  value  of  P,  which 
maximum  determines  (Art.  319.)  the  thrust  of  the  earth 
upon  the  portion  AX  of  the  wall.  To  obtain  this  maximum 
value  of  P  by  substitution  in  equation  (441),  let  it  be 
observed  that 

cos.  (*  +  <p)_l—  tan.  i  tan.  9    /cos.  <p  \ 
sin.  (i  +  $)~~  (tan-  i+tan.  /3)    \cos.  j8'/ 


1—  tan.  i  tan.  9=1  +  tan.  /3  tan.  9—  tan.  9  (tan.  /3— 

tan.  a2)*(tan.  /3-fcot.  9)*, 

=tan.  9  (tan.  /3  +  cot.  9)M(tan-  1s  +  cot.  9)*—  (tan.  /3—  tan. 
tan.  i+tan.  ^=(tan.  /3  +  cot.  9)f(tan.  /3—  tan.  aa)*; 

cos.  (t.  +  9)     sin.  9  j   /  tan.  /3  -f  cot.  9  \  *          ) 
•'"sin.  (t  +  /3=cos.  /3  (  \tan.  /3—  tan.  a)  "         )  ' 


Also  tan.  i  +  tan.  a2=(tan.  /3-fcot.  9)*(tan.  /3— 

tan.  aa)}—  (tan.  /3—  tan.  aa) 
—(tan.  /3—  tan.  a,)*  {(tan.  ^  +  cot.  9)*—  (tan.  p—  tan. 


^y-Ktan.  /3  +  cot.  9)*-(tan.  £-tan.  «,)*{  a  ; 

COS.  /3 


424 


RE  VETE3IENTS . 


which  expression  may  be  placed  under  the  following  form, 
better  adapted  to  logarithmic  calculation, 

P_i      •  sin-  9  \  /cos.  (ff-<p)\  *_  /sin.  (/3-q,)\  *  )  8  . 
~tM/1    cos.  2/3  (  \      sin.  9      /       \     cos.  aa     /    f 

or  substituting  for  (3  its  value  a3+<P+<p2? 

P=l      ^sin-<P        J    /COS.(aa  +  9,)\*_ 
~2  \      sin.  9       / 

-       >( 


By  a  comparison  of  this  equation  with  equation  (427)  it 
is  apparent,  that  the  pressure  of  a  mass  of  earth  upon  a 
revetement  wall,  under  the  supposed  conditions,  is  identical 
with  that  which  it  would  produce  if  it  were  perfectly  fluid, 
provided  that  the  weight  of  each  cubic  foot  of  that  fluid  had 
a  value  represented  by  the  coefficient  of  -Jar2  in  the  above 
equation  ;  so  that  the  conditions  of  the  stability  of  such  a 
revetement  wall  are  identical  (this  value  being  supposed) 
with  the  conditions  of  the  stability  of  a  wall  sustaining  the 
pressure  of  a  fluid,  except  that  the  pressure  of  the  earth  is 
not  exerted  upon  the  wall  in  a  direction  perpendicular  to  its 
surface,  as  that  of  a  fluid  is,  but  in  a  direction  inclined  to 
the  perpendicular  at  a  given  angle,  namely,  the  limiting 
angle  of  resistance. 


328.  THE  PRESSURE  OF  EARTH  WHICH  SURMOUNTS  A  REVETE- 

MENT WALL   AND   SLOPES   TO   ITS    SUMMIT. 


Hitherto  we  have  supposed  the  surface  of  the  earth  whose 


ippose 

vated  above  the  summit  of  the  wall, 
and  to  descend  to  it  by  the  natural 
slope ;  the  wall  is  then  said  to  be 
surcharged,  or  to  carry  a  parapet. 
Let  EF  represent  the  natural  slope 
of  the  earth,  FY  its  horizontal  sur- 
face, BX  any  portion  of  the  internal 
face  or  intrados  of  the  wall,  P  the 
horizontal  pressure  just  necessary 


REVETEMENTS.  425 

f 

to  support  the  mass  of  earth  HXYF,  whose  weight  is  W, 
upon  the  inclined  plane  XY.  Produce  XB  and  YF  to  meet 
in  A,  and  let  AX^a?,  AH=£,  AXY=u,  ^=  weight  of  each 
cubic  foot  of  the  earth,  9  the  natural  slope  of  its  surface 
FE.  Now  it  may  be  shown,  precisely  by  the  same  reason- 
ing as  before,  that  the  actual  pressure  of  the  earth  upon  the 
portion  BX  of  the  wall  is  represented  by  that  value  of  P 
which  is  a  maximuni  in  respect  to  the  variable  i  ;  moreover, 
that  the  relation  of  P  and  i  is  expressed  by  the  function  P 
=  W  cot.  0+9);  where  W^f^area  HXYF)=^(AXY— 
—  |-cacot.  9); 


/.P—  ^(a?8  tan.  i—  <?  cot.  9)  cot.  (4+9)  .....  (4M). 
Expanding  cot.  (^+9), 

P__JL   (#a  tan.  i—  c*  cot.  9)  (1—  tan,  i  tan.  9) 
~"*V  tan.  i+  tan.  9 

To  facilitate  the  differentiation  of  this  function,  let 
tan.  *-f  tan.  9  be  represented  by  z,  and  let  it  be  observed 
that  whatever  conditions  determine  the  maximum  value  of  P 
in  respect  to  z  determine  also  its  maximum  value  in  respect 
to  i.*  Then  tan.  i=z—  tan.  9  ;  therefore  1  —  tan.  i  tan.  9= 
1—z  tan.  9  +  tan.  29=—  z  tan.  9  +  sec.  "9.  Also,  a?a  tan.  <— 
c*  cot.  q>=a?z—  (a?  tan.  9+ca  cot.  9). 

Substituting  these  values  in  the  preceding  expression  for 
P,  and  reducing, 

j  («?"  tan.  9  +  c2  cot.  9)  sec.  '9 

P=i^i  \  —z®  ^n.  9—  -  -  -  -  l  -  -+ 


.....  (445). 

dl*  (  (a;8  tan.  9  +  ca  cot.  9)  sec.  *9  ) 

a  -p-  -  \  , 


^_      dP    dPdz         ,    d*P    rfTP/&\«    rfP<f»«  dz  ,      t 

*  For  -r=-T-  -j-.  and  — — =_ _(  —    -f— —  — -  ;  now  — -  =sec.  *f.  there- 
dt     dz  dt  dt1      dz*\dt  )       dz  dt*  dt 

7T>  /JT> 

fore  -j-  =  -=-  sec.  3i;  and  for  all  values  of  i  less  than  -,  sec.  2i  has  a  finite 
dt      dz  2' 

value,  so  that  -3-  =  0  when  — =0. 
<w  efe 

</P         <iaP     d*P  idz\*  d*P 

When,  moreover,  -T-=O,  -TT=-rr  ( -7- )  5  so  tnat»  when  -r-r-  is  negative. 
az          at        dz    \di  /  dz 

-; —  is  also  negative. 


426  BEVETEMENTS. 

<?  P_         («j*  tan,  9  +  c9  cot.  9)  sec.  '9 

~~ 


The  first  condition  of  a  maximum  is  therefore  satisfied  by 
the  equation 

(a;2  tan.  9  +<?8  cot.  9)  sec.  "9     A  ,.,„, 

—  a?atan.9  +  -  -  -9  -         -=0  .  .  .  .  (446); 

or,  solving  this  equation  in  respect  to  z,  and  reducing,  it  is 
satisfied  by  the  equation 


/  <?2  \* 

'<=  ±  I  sec.  9  +  — a  cosec.  9 1  . 


E"ow  the  second  condition  of  a  maximum  is  evidently 
satisfied  by  any  positive  value  of  z,  and  therefore  by  the 
positive  root  of  this  equation.  Taking,  therefore,  the  posi- 
tive sign,  substituting  for  z  its  value,  and  transposing, 

/  c2  \* 

tan.  i=  I  sec.  \  -f  —  cosec.  2<p  1  —tan.  9  .....  (447)  ; 

which  equation  determines  the  tangent  of  the  inclination 
AXY  to  the  vertical,  of  the  base  XY  of  that  wedge-like 
mass  of  earth  HXYF,  whose  pressure  is  borne  by  the  sur- 
face BX  of  the  wall.  To  determine  the  actual  pressure 
upon  the  wall,  this  value  of  tan.  i  must  be  substituted  in  the 
expression  for  P  (equation  445).  Now  the  two  first  terms 
of  the  expression  within  the  brackets  in  the  second  member 
of  that  equation  may  be  placed  under  the  form 


( 
—  z  I 


(a?2  tan.  9+c2  cot.  9)  sec.  3<p 
a?2  tan.   9  +  ^  -  --  "  - 


But  it  appears  by  equation  (446)  that  the  two  terms  which 
compose  this  expression  are  equal,  so  that  the  expression  is 
equivalent  to  —  2^»2  tan.  9  ;  or,  substituting  for  the  value  of 

Z)  to  —  2ft2  tan.  9  (sec.  a9+-j  cosec.  *<p)*,  or  to  —  2a?  sec.  9 
(a?2  tan.  29  +  c2)  *  .  Substituting  in  equation  (445), 

P—  ^{—  2a?sec.  9(#3tan.a9  +  ca)*  +(a?atan.  29  +  ca)+arl  sec.  '9} 
;.P=lfi.l{jB  sec.  9—  (#2  tan.  29  +  <?2)^2  .....  (448); 


by  which  expression  is  determined  the  actual  pressure  upon 
a  portion  of  the  wall,  the  distance  of  whose  lowest  point 
from  A  is  represented  by  x. 


427 


329.  The  conditions  necessary  that  a  revetement  wall  carry- 
ing a  parapet  may  not  be  overthrown  by  the  slipping  of 
any  course  of  stones  on  the  subjacent  course. 

Let  <pj  represent  the  limiting  angle  of  the  resistance  of  the 
stones  of  the  wall  upon  one  another ;  and  let  OQ  represent 
the  direction  of  the  resultant  pressure 
on  the  course  XZ.  The  proposed 
conditions  are  then  involved  (Art. 
141.)  in  the  inequality  91>QOM,  or 
tan.  9j  >  tan.  QOM,  or  tan.  9,  > 

T^> — ^tnr-^ov*;  or  substituting 
OS  weight  of  BZ* 

for  P  its  value  (equation  448),  and 
£fx(2#a?-fa?2  tan.  a)  for  the  weight  of 
BZ,  it  appears  that  the  proposed 
conditions  are  determined  by  the 
inequality 


tan. 


tan.  a 


330. 


of  resistance  in  a  revetement  wall  carrying  a 
parapet. 


Let  OT  be  taken  to  represent  the  pressure  P,  and  OS  the 
weight  of  BZ.  Complete  the  parallelogram  ST,  and  pro- 
duce its  diagonal  OR  to  Q  ;  then  will  Q  be  a  point  in  the 
line  of  resistance.  Let  AX=#,  QX=£/,  AB=5,  AP=X, 


=X,  W  =  weight  of  BZf.     By  similar  triangles,     r?= 


g    ;  but  QM=(y-X),  OM=aj-X,  KS=P,  OS= 
y-X      P  Wx+Paj-PX 

~ 


Now  the  value  of  X  is  determined  from  equation  (414),  by 

*  The  influence,  upon  the  equilibrium  of  the  wall,  of  'the  small  portion  of 
earth  BHE  is  neglected  in  this  and  the  subsequent  computation. 

f  The  influence  of  the  weight  of  the  small  mass  of  earth  BEH  which  rests 
on  the  summit  of  the  wall  is  here  again  neglected. 


428 


REVETEMENTS. 


substituting  in  that  equation  (x—  1)  for  c:  whence  we  obtain, 
observing  that  tan.  aa=0,  and  substituting  a  for  a1? 


.  _i(#—  &)a  tan.  aa  +  #(#—£)  tan.  a 
(a?—  5)tan.aH-2# 

Also    W=JKa-5){(a-&)tan.a  +  2a{  .....  (451); 
/.  WX=  Jf*(»—  &)  $(o?—  &)a  tan.  2a  +  a(aj—  5)  tan.  a 


It  remains,  therefore,  only  to  determine  the  value  of  the 
term  P  .  X.  Now  it  is  evident  (Art.  16.)  that  the  product 
P  .  X  is  equal  to  the  sum  of  the  moments  of  the  pressures 
upon  the  elementary  surfaces  which  compose  the  whole  sur- 
face BX.  But  the  pressure  upon  any  such  elementary  sur- 
face, whose  distance  from  A  is  a?,  is  evidently  represented 

by  -J-AX*  ;  its  moment  is  therefore  represented  by  -r-ajAo?, 
and  the  sum  of  the  moments  of  all  such  elementary  pressures 
by  2—  -ccAaj,  or  when  AX  is  infinitely  small,  by 

/  -T-xdx  ;  therefore  P  .  X=  / 

b  b 

But  differentiating  equation  (448), 

2  <p 


Performing  the  actual  multiplication  of  the  factors  in  the 
second  member  of  this  equation,  observing  that  -^  --  ~  -^-yr 

3     ,  ,  ca 

=(x  tan.V  +  ^y—  /—  rr  -  a    ,    av,  and  re- 
y         2a        aJ 


ducing  we  obtain 

*  P  being  a  function  of  #,  let  it  be  represented  by  f(x);  then  will  f(x)  repre- 
Bent  the  pressure  upon  a  portion  of  the  surface  BX  terminated  at  the  distance 
x  from  A,  and  /(a-f-Az)  that  upon  a  portion  terminated  at  the  distance  x-\-Ax  ; 
therefore  /(  x  -f-Aic)—  fx  will  represent  the  pressure  upon  the  small  element  Aa? 
of  the  surface  included  between  these  two  distances.  But  by  Taylor's  theorem, 

/(ar-f-Aa;)—  fx=  -•-  As  -f-  —  —  ^-~-  -f,  &c.  ;  therefore,  neglecting  terms  ii> 
ux  dx     1*2 

volving  powers  of  A#  above  the  first,  pressure  on  element  =  -j-A*. 


THE  ABOH.  429 


dx~ 

<?  sec.  9 


Multiplying  this  equation  by  a?,  and  integrating  between  the 
limits  6  and  a?, 

'i(sec.a  9+tan.2  9)(a?8— 58)— |  sec.  9  cot.  '9  |(a?a  tan.a9 
P.X^^-I          +ca)f — (V2  tan.a  9  +  <?a)f}  +c2  sec.  9  cot.2  9 

....  (452). 


This  value  of  P  .  X  being  substituted  in  equation  (450), 
and  the  values  of  W\  W,  P,  from  equations  (448)  and 
(451),  the  line  of  resistance  to  the  revetement  wall  will  be 
determined,  and  thence  all  the  conditions  of  its  stability 
may  be  found  as  before.* 


THE  ARCH. 

331.  Each  of  the  structures,  the  conditions  of  whose  sta- 
bility (considered  as  a  system  of  bodies  in  contact),  have 
hitherto  been  discussed,  whatever  may  have  been  the  pres- 
sures supposed  to  be  insistent  upon  it,  has  been  supposed  to 
rest  ultimately  upon  a  single  resisting  surface,  the  resultant 
of  the  resistances  on  the  different  elements  of  which  was  at 
once  determined  in  magnitude  and  direction  by  the  resultant 
of  the  given  insistent  pressures!  being  equal  and  opposite 
to  that  resultant. 

The  arch  is  a  system  of  bodies  in  contact  which  reposes 
ultimately  upon  two  resisting  surfaces  called  its  abutments. 
The  resistances  of  these  surfaces  are  in  equilibrium  with  the 

*  The  limits  which  the  author  has  in  this  work  imposed  upon  himself  do  not 
leave  him  space  to  enter  further  upon  the  discussion  of  this  case  of  the 
revetement  wall,  the  application  of  which  to  the  theory  of  fortification  is  so 
direct  and  obvious.  The  reader  desirous  of  further  information  is  referred  to 
the  treatise  of  M.  Poncelet,  entitled  "  Memoire  sur  la  Stabilite  des  Revete- 
ments  et  do  lours  Fondations."  He  will  there  find  the  subject  developed  in  all 
ith  practical  relations,  and  treated  with  the  accustomed  originality  and  power 
of  that  illustrious  author.  The  above  method  of  investigation  has  nothing  in 
common  with  the  method  adopted  by  M.  Poncelet  except  Coulomb's  principle 
of  the  wedge  of  maximum  pressure. 

f  The  weight  of  the  structure  itself  is  supposed  to  be  included  among  these 
pressures.  . 


430  THE   PRINCIPLE   OF   LEAST   RESISTANCE. 

given  pressures  insistent  upon  the  arch  (inclusive  of  its 
weight),  but  the  direction  and  amount  of  the  resultant  pres- 
sure upon  each  surface  is  dependent  upon  the  unknown 
resistance  of  the  opposite  surface ;  and  thus  the  general 
method  applicable  to  the  determination  of  the  fine  of 
resistance,  and  thence  of  the  conditions  of  stability,  in  that 
large  class  of  structures  which  repose  on  a  single  resisting 
surface,  fails  in  the  case  of  the  arch. 


332.  THE  PRINCIPLE  OF  LEAST  RESISTANCE. 

If  there  ~be  a  system  of  pressures  in  equilibrium  amona  which 
are  a  given  number  of  resistances,  then  is  each  of  these  a 
minimum,  subject  to  the  conditions  imposed  by  the  equili- 
brium of  the  whole* 

Let  the  pressures  of  the  system,  which  are  not  resistances, 
be  represented  by  A,  and  the  resistances  by  B  ;  also  let  any 
other  system  of  pressures  which  may  be  made  to  replace  the 
pressures  B  and  sustain  A,  be  represented  by  C. 

Suppose  the  system  B  to  be  replaced  by  C ;  then  it  is 
apparent  that  each  pressure  of  the  system  C  is  equal  to  the 
pressure  propagated  to  its  point  of  application  from  the 
pressures  of  the  system  A ;  or  it  is  equal  to  that  pressure, 
together  with  the  pressure  so  propagated  to  it  from  the 
other  pressures  of  the  system  C.  In  the  former  case  it  is 
identical  with  one  of  the  resistances  of  the  system  B  ;  in  the 
latter  case  it  is  greater  than  it.  Hence,  therefore,  it  appears 
that  each  pressure  of  the  system  B  is  a  minimum,  subject 
to  the  conditions  imposed  by  the  equilibrium  of  the  whole. 

If  the  resultant  of  the  pressures  applied  to  a  body,  other 
than  the  resistances,  be  taken,  it  is  evident  from  the  above 
that  these  resistances  are  the  least  possible  so  as  to  sustain 
that  resultant ;  and  therefore  that  if  each  resisting  point  be 
capable  of  supplying  its  resistance  in  any  direction,  then  are 
all  the  resistances  parallel  to  one  another  and  to  the  result- 
ant of  the  other  pressures  applied  to  the  body. 

*  The  principle  of  least  resistance  was  first  published  by  the  author  of  thia 
work  in  the  Philosophical  Magazine  for  October,  1833. 


THE   ARCH.  431 

333.  Of  oil  the  pressures  which  can  le  applied  to  the  highest 
voussoir  of  a  semi-arch,  different  in  their  amounts  and 
points  of  application,  hut  all  consistent  with  the  equili- 
orium  of  the  semi-arch,  that  which  it  would  sustain  from 
the  pressure  of  an  opposite  and  equal  semi-arch  is  the  least. 

Let  EB  represent  the  surface  by  which  an  arch  rests  upon 


either  of  its  abutments ;  then  are  the  resistances  upon  the 
different  points  of  that  surface  (Art.  331.)  the  least  pressures, 
which,  being  applied  to  those  points,  are  consistent  with  the 
equilibrium  of  the  arch.  They  are,  moreover,  parallel  to  one 
another  :  their  resultant  is  therefore  the  least  single  pressure, 
which,  being  applied  to  the  surface  EB,  would  be  sufficient 
to  maintain  the  equilibrium  of  the  arch,  if  the  abutment  were 
removed. 

Now,  if  this  resultant  be  resolved  vertically  and  horizon- 
tally, its  component  in  a  vertical  direction  will  evidently  be 
equal  to  the  weight  of  the  semi-arch  :  it  is  therefore  given  in 
amount.  In  order  that  the  resultant  may  be  a  minimum,  its 
vertical  component  being  thus  given,  it  is  therefore  necessary 
that  its  horizontal  component  should  be  a  minimum ;  but 
this  horizontal  component  of  the  resistance  upon  the  abut- 
ment is  evidently  equal  to  the  pressure  P  of  the  opposite 
semi- arch  upon  its  key-stone  :  that  pressure  is  therefore  a 
minimum;  or,  if  the  semi-arches  be  equal  in  every  respect, 
it  is  the  least  pressure  which,  being  applied  to  the  side  of  the 
key-stone,  would  be  sufficient  to  support  either  semi-arch ; 
which  was  to  be  proved. 

The  following  proof  of  this  property  may  be  more  intelli- 
gible to  some  readers  than  the  preceding.  It  is  independent 
of  the  more  general  demonstration  of  the  principle  of  least 
resistance.* 

*  See  Memoir  by  the  author  of  this  work  in  Mr.  Harm's  "  Treatise  on  the 
Theory  of  Bridges,"  p.  10. 


432 


THE   AKCH. 


The  pressure  which  an  opposite  semi-arch  would  produce 
upon  the  side  AD  of  the  key-stone,  is  equal  to  the  tendency 
of  that  semi-arch  to  revolve  forwards  upon  the  inferior  edges 
of  one  or  more  of  its  voussoirs.  Now  this  tendency  to  motion 
is  evidently  equal  to  the  least  force  which  would  support  the 
opposite  semi-arch.  If  the  arches  be  equal  and  equally 
loaded,  it  is  therefore  equal  to  the  least  force  which  would 
support  the  semi-arch  ABED. 


334:.  GENERAL  CONDITIONS  OF  THE  STABILITY  OF  AN 


Suppose  the  mass  ABDO  to  be  acted  upon  by  any  number 

of  pressures,  among  which 
is  the  pressure  Q,  being  the 
resultant  of  certain  resist- 
ances, supplied  by  different 
points  in  a  surface  BD  ; 
common  to  the  mass  arid  to 
an  immoveable  obstacle 
BE. 

Now  it  is  clear  that  un- 
der these  circumstances  we 
may  vary  the  pressure  P, 
both  as  to  its  amount,  di- 
rection, and  point  of  appli- 
cation in  AC,  without  disturbing  the  equilibrium,  provided 
only  the  form  and  direction  of  the  line  of  resistance  continue 
to  satisfy  the  conditions  imposed  by  the  equilibrium  of  the 
system. 

These  have  been  shown  (Art.  283)  to  be  the  following  :  — 
that  it  no  where  cut  the  surface  of  the  mass,  except  at  P, 
and  within  the  space  BD  ;  and  that  the  resultant  pressure 
upon  no  section  MN  of  the  mass,  or  the  common  surface  BD 
of  the  mass  and  obstacle,  be  inclined  to  the  perpendicular  to 
that  surface,  at  an  angle  greater  than  the  limiting  angle  of 
resistance. 

Thus,  varying  the  pressure  P,  we  may  destroy  the  equi- 
librium, either,  first,  by  causing  the  resultant  pressure  to 
take  a  direction  without  the  limits  prescribed  by  the  resist- 
ance of  any  section  MN  through  which  it  passes,  that  is, 
without  the  cone  of  resistance  at  the  point  where  it  inter- 

*  Theoretical  and  Practical  Treatise  on  Bridges,  vol.  i.  ;  Memoir  by  the  au- 
thor of  this  work,  p.  11. 


THE   ARCH.  433 

sects  that  surface ;  or,  secondly,  by  causing  the  point  Q  to 
fall  without  the  surface  BD,  in  which  case  no  resistance  can 
be  opposed  to  the  resultant  force  acting  in  that  point ;  or, 
thirdly,  the  point  Q  lying  within  the  surface  BD,  we  may 
destroy  the  equilibrium  by  causing  the  line  of  resistance  to 
cut  the  surface  of  the  mass  somewhere  between  that  point 
and  P. 

Let  us  suppose  the  limits  of  the  variation  of  P,  within 
which  the  first  two  conditions  are  satisfied,  to  be  known  ;  and 
varying  it,  within  those  limits,  let  us  consider  what  may  be 
its  least  and  greatest  values  so-  as-  to  satisfy  the  third  condition,, 

Let  P  act  at  a  given  point  in  AC,  and  in  a  given  direc- 
tion. It  is  evident  that  by  diminishing  it  under  these 
circumstances  the  line  of  resistance  will  be- made  continually 
to  assume  more  nearly  that  direction  which  it  would  have 
if  P  were  entirely  removed. 

Provided,  then,  that  if  P  were  thus  removed,  the  line  of 
resistance  would  cut  the  surface, — that  is,  provided  the 
force  P  be  necessary  to  the  equilibrium, — it  follows  that  by 
diminishing  it  we  may  vary  the  direction  and  curvature  of 
the  line  of  resistance,  until  we  at  length  make  it  touch  some 
point  or  other  in  the  surface  of  the  mass. 

And  this  is  the  limit ;  for  if  the  diminution  be  carried 
further,  it  will  cut  the  surface,  and  the  equilibrium  will  be 
destroyed.  It  appears,  then,  that  under  the  circumstances 
supposed,  when  P,  acting  at  a  given  point  and  in  a  given 
direction,  is  the  least  possible,  the  line  of  resistance  touches 
the  interior  surface  or  intrados  of  the.  mass. 

In  the  same  manner  it  may  be  shown  that  when  it  is  the 
greatest  possible,  the  line  of  resistance  touches  the  exterior 
surface  or  extrados  of  the  mass. 

The  direction  and  point  of  application  of  P  in  AC  have 
here  been  supposed  to  be  given ;  but  by  varying  this  direc- 
tion and  point  of  application,  the  contact  of  the  line  of 
resistance  with  the  intrados  of  the  arch  may  be  made  to 
take  place  in  an  infinite  variety  of  different  points,  and  each 
such  variation  supplies  a  new  value  of  P.  Among  these, 
therefore,  it  remains  to  seek  the  absolute  maximum  and 
minimum  values  of  that  pressure. 

In  respect  to  the  direction  of  the  pressure  P,  or  its  incli- 
nation to  AC,  it  is  at  once  apparent  that  the  least  value  of 
that  pressure  is  obtained,  whatever  be  its  point  of  applica- 
tion, when  it  is  horizontal. 

There  remain,  then,  two  conditions  to  which  P  is  to  be 
subjected,  and  which  involve  its  condition  of  a  minimum. 

28 


434  THE  ARCH. 


The  first  is,  that  its  amount  shall  ~be  such  as  will  give  to  the 
line  of  resistance  a  point  of  contact  with  the  intrados  ;  the 
second,  that  its  point  of  application  in  the  key-stone  AC 
shall  be  such  as  to  give  it  the  least  value  which  it  can  receive, 
subject  to  the  first  condition. 


335.  PRACTICAL  CONDITIONS  OF  THE  STABILITY  or  AN  AECH 

OF    TJNCEMENTED    STONES. 

The  condition,  however,  that  the  resultant  pressure  upon 
the  key-stone  is  subject,  in  respect  to  the  position  of  its 
point  o'f  application  on  the  key-stone,  to  the  condition  of  a 
minimum,  is  dependent  upon  hypothetical  qualities  of  the 
masonry.  It  supposes  an  unyielding  material  for  the  arch- 
stones,  and  a  mathematical  adjustment  of  their  surfaces. 
These  have  no  existence  in  the  uncemented  arch.  On  the 
striking  of  the  centres  the  arch  invariably  sinks  at  the 
crown,  its  voussoirs  there  slightly  opening  at  their  lower 
edges,  and  pressing  upon  one  another  exclusively  by  their 
upper  edges.  Practically,  the  line  of  resistance  then,  in  an 
arch  of  uncemented  stones,  touches  the  extrados  at  the  crown ; 
so  that  only  the  iirst  of  the  two  conditions  of  the  minimum 
stated  above  actually  obtains :  that,  namely,  which  gives  to 
the  line  of  resistance  a  contact  with  the  intrados  of  the 
arch.  This  condition  being  assumed,  all  consideration  of 
the  yielding  quality  of  the  material  of  the  arch  and  its 
abutments  is  eliminated. 

The  form  of  the  solid  has  hitherto  been  assumed  to  be 
given,  together  with  the  positions  of  the  different  sections 
made  through  it ;  and  the  forms  of  its  lines  of  resistance  and 
pressure,  and  their  directions  through  its  mass  have  thence 
been  determined. 

It  is  manifest  that  the  converse  of  this  operation  is  pos- 
sible. 

Having  given  the  form  and  position  of  the  line  of  resist- 
ance or  of  pressure,  and  the  positions  of  the  different  sections 
to  be  made  through  the  mass,  it  may,  for  instance,  be 
inquired  what  form  these  conditions  impose  upon  the  surface 
which  bounds  it. 

Or  the  direction  of  the  line  of  resistance  or  pressure  and 
the  form  of  the  bounding  surface  may  be  subjected  to  certain 
conditions  not  absolutely  determining  either. 

If,  for  instance,  the  form  of  the  intrados  of  an  arch  be 
given,  and  the  direction  of  the  intersecting  plane  be  always 


THE   ARCH.  435 

perpendicular  to  it,  and  if  the  line  of  pressure  be  supposed 
to  intersect  this  plane  always  at  the  same  given  angle  with 
the  perpendicular  to  it,  so  that  the  tendency  of  the  pressure 
to  thrust  each  from  its  place  may  be  the  same,  we  may 
determine  what,  under  these  circumstances,  must  be  the 
extrados  of  the  arch. 

If  this  angle  equal  constantly  the  limiting  angle  of  resist- 
ance, the  arch  is  in  a  state  bordering  upon  motion,  each 
voussoir  being  upon  the  point  of  slipping  downwards,  or  up- 
wards, according  as  the  constant  angle  is  measured  above  or 
below  the  perpendicular  to  the  surface  of  the  voussoir. 

The  systems  of  voussoirs  which  satisfy  these  two  con- 
ditions are  the  greatest  and  least  possible. 

If  the  constant  angle  be  zero,  the  line  of  pressure  being 
every  where  perpendicular  to  the  joints  of  the  voussoirs,  the 
arch  would  stand  even  if  there  were  no  friction  of  their  sur- 
faces. It  is  then  technically  said  to  be  equilibriated ;  and 
the  equilibrium  of  the  arch,  according  to  this  single  con- 
dition, constituted  the  theory  of  the  arch  so  long  in  vogue, 
and  so  well  known  from  the  works  of  Emerson,  Hutton,  and 
"Whewell.  It  is  impossible  to  conceive  any  arrangement  of 
the  parts  of  an  arch  by  which  its  stability  can  be  more 
effectually  secured,  so  far  as  the  tendency  of  its  voussoirs  to 
slide  upon  one  another  is  concerned:  there  is,  however, 
probably,  no  practical  case  in  which  this  tendency  really 
affects  the  equilibrium.  So  great  is  the  limiting  angle  of 
resistance  in  respect  to  all  the  kinds  of  stone  used  in  the 
construction  of  arches,  that  it  would  perhaps  be  difficult  to 
construct  an  arch,  the  resultant  pressure  upon  any  of  the 
joints  of  which  above  the  springing  should  lie  without  this 
angle,  or  which  should  yield  by  the  slipping  of  any  of  its 
voussoirs. 

Traced  to  the  abutment  of  the  arch,  the  line  of  resistance 
ascertains  the  point  where  the  direction  of  the  resultant 
pressure  intersects  it,  and  the  line  of  pressure  determines  the 
inclination  to  the  vertical  of  that  resultant  ;*  these  elements 
determine  all  the  conditions  of  the  equilibrium  of  the  abut- 
ments, and  therefore  of  the  whole  structure ;  they  associate 
themselves  directly  with  the  conditions  of  the  loading  of  the 
arch,  and  enable  us  so  to  distribute  it  as  to  throw  the  points 
of  rupture  into  any  given  position  on  the  intrados,  and  give 
to  the  line  of  resistance  any  direction  which  shall  best  con- 

*  The  inclination  of  the  resultant  pressure  at  the  springing  to  the  vertical 
may  be  determined  independently  of  the  line  of  pressure,  as  will  hereafter  be 
shown 


4:36  THE  LINE   OF  RESISTANCE   IN   THE   AKCH. 

duce  to  the  stability  of  the  structure ;  from  known  ^dimen- 
sions,  and  a  known  loading  of  the  arch,  they  determine  the 
dimensions  of  piers  which  will  support  it ;  or  conversely, 
from  known  dimensions  of  the  piers  they  ascertain  the 
dimensions  and  loading  of  the  arch,  which  may  safely  be 
made  to  span  the  space  between  them. 


336.  To  DETERMINE  THE  LINE  OF  RESISTANCE  IN  AN  ARCH 
WHOSE  INTRADOS  18  A  CIRCLE,  AND  WHOSE  LOAD  IS  COL- 
LECTED OVER  TWO  POINTS  OF  ITS  EXTRADOS  SYMMETRICALLY 
PLACED  IN  RESPECT  TO  THE  CROWN  OF  THE  ARCH. 

Let  ADBF  represent  any  portion  of  such  an  arch,  P  a 

pressure  applied  at  its  extreme 
voussoir,  and  X  and  Y  the  ho- 
rizontal and  vertical  compo- 
nents of  any  pressure  borne 
upon  the  portion  DT  of  its  ex- 
trados,  or  of  the  resultant  of 
any  number  of  such  pressures  ; 
let,  moreover,  the  co-ordinates, 
from  the  centre  C,  of  the  point 
of  application  of  this  pressure, 
or  of  this  resultant  pressure,  be 
x  and  y. 

Let  the  horizontal  force  P 
be  applied  in  AD  at  a  vertical  distance  p  from  C  ;  also  let 
CT  represent  any  plane  which,  passing  through  0,  intersects 
the  arch  in  a  direction  parallel  to  the  joints  of  its  voussoirs. 
Let  this  plane  be  intersected  by  the  resultant  of  the  pres- 
sures applied  to  the  mass  ASTD  in  R.  These  pressures  are 
the  weight  of  the  mass  ASTD,  the  load  X  and  Y,  and  the 
pressure  P.  Now  if  pressures  equal  and  parallel  to  these, 
but  in  opposite  directions,  were  applied  at  K,  they  would  of 
themselves  support  the  mass,  and  the  whole  of  the  subjacent 
mass  TSB  might  be  removed  without  affecting  the  equili- 
brium. (Art.  8.)  Imagine  this  to  be  done  ;  call  M  the  weight 
of  the  mass  ASTD,  and  k  the  horizontal  distance  of  its  cen- 
tre of  gravity  from  C,  and  let  CR  be  represented  by  p,  and 
the  angle  ECS  by  0,  then  the  perpendicular  distances  from 
C  of  the  pressures  M+ Y  and  P— X,  imagined  to  be  applied 
to  R,  are  p  sin.  6  and  p  cos.  6 ;  therefore  by  the  condition  of 
the  equality  of  moments, 


THE  ANGLE  OF  KUPTUKE  IN  THE  ARCH. 


437 


(M+Y)  p  sin.  d  +  (P— X)  p  cos.a=MA+ Y»-Xy+Pp ; 


P~~ 


(M+Y)  sin.d-f  (P-X)  cos.  6 


.  .  .  (453), 


which  is  the  equation  to  the  line  of  resistance. 

M  and  h  are  given  functions  of  & ;  as  also  are  X  and  Y,  if 

the  pressure  of  the  load  extend  continuously  over  the  surface 

of  the  extrados  from  D  to  T. 

It  remains  from  this  equation 
to  determine  the  pressure  P,  be- 
ing that  supplied  by  the  opposite 
semi-arch.  As  the  simplest  case, 
let  all  the  voussoirs  of  the  arch 
be  of  the  same  depth,  and  let  the 
inclination  ECP  of  the  first  joint 
of  the  semi-arch  to  the  vertical  be 
represented  by  0,  and  the  radii 
of  the  extrados  and  intrados  by 
R  and  r.  Then,  by  the  known 
principles  of  statics.* 

R      e 
=  /  J  r3  sin.  &d&dr=— i(Rs— ^(cos.  6— cos.  0) ; 

r         6 

also,  M=-KRf— r>)(a— e) ; 

/.  p  $(R8-raX0-0)  sin.  6+ Y  sin.  d-X  cos.  d+P  cos.  A]  = 

which  is  the  general  equation  to  the  line  of  resistance. 


THE  ANGLE  OF  RUPTURE. 

337.  At  the  points  of  rupture  the  line  of  resistance  meets 
the  intrados,  so  that  there  p=r  :  if  then  Y  be  the  correspond- 
ing value  of  d, 

r  $(R2— r*)(v— ©)  sin.  Y  +  Y  sin.  Y— X  cos.  Y  +  P  cos.  Y}  =- 
3— /)(cos.  0— cos.¥)+Y#— Xy  +  P^? (455). 


See  Note  1  at  end  of  PART  IV.— ED. 


438  THE   ANGLE   OF   KUPTUKE 

Also  at  the  points  of  rupture  the  line  of  resistance  toucJws 
the  intrados,  so  that  there  -1=-^=®  5   assuming  then,  tu 

OJO        U/a 

simplify  the  results,  that  the  pressure  of  the  load  is  wholly 
in  a  vertical  direction,  so  that  X=0,  and  that  it  is  collected 

/7~V 

over  a  single  point  of  the  extrados,  so  that  —  =0,  and  dif- 

ferentiating equation  (454),  and  assuming  -^=0,  when  d=Y 
and  p=r,  we  obtain 

r  {i(K2-r2)  (v—  0)  cos.  Y  +  £  (Ea—  ra)  sin.  Y  + 

Y  cos.  Y—  P  sin.  ^J  =4(K3-O  sin.  Y  ; 
hence,  assuming  R^T1  (1+a), 


(456). 

Eliminating  (Y—  0)  between  equations  (455)  and  (456),  we 
have 


(457). 


Eliminating  P   between    equations    (455)    and    (456),   and 
reducing. 


sin.  Y—  {(«+«'+$«')  cos.  ©—  (Ja'+a)  cos.Yj  gin.Y 

.....  (458). 


*  This  equation  might  have  been  obtained  by  differentiating  equation  (454) 

7T> 

in  respect  to  P  and  6,  and  assuming  -  —  =  0  when  r  and  ¥  are  substituted  for 
p  and  0;  for  if  that  equation  be  represented  by  w—  0,  u  being  a  function  of 

is 


therefore  obtained,  whether  we  assume-—  =  0,  or-—  =0,  which  last  supposi- 

dv  da 

tion  is  that  made  in  equation  (456),  whence  equation  (458)  has  resulted.     The 


IN  THE  AKCH. 


439 


Let 

(1+X)  cos.  ©. 
value  of  —  , 

T 


;   therefore     -= 
Substituting  this 


cos. 


jl-(l+X)  cos.  ©cos.  ¥}(¥-- 


-1      =(iaa+  a) 
.¥—  cos.©)sin.Y  I 


by  which  equation  the  angle  of  rupture  Y  is  determined. 

If  the  arch  be  a  continuous  segment  the  joint  AD  is  ver- 
tically above  the  centre,  and  CD  coinciding  with  CE,  ©=0; 
if  it  be  a  broken  segment,  as  in  the  Gothic  arch,  ©  has  a 
given  value  determined  by  the  character  of  the  arch.  In  the 
pure  or  equilateral  Gothic  arch,  0  =  30°.  Assuming  0=0, 
and  reducing, 


(460.) 


It  may  easily  be  shown  that  as  Y  increases  in  this  equa- 
tion, Y  increases,  and  conversely  ;  so  that  as  the  load  is 
increased,  the  points  of  rupture  descend.  When  Y=0,  or 
there  is  no  load  upon  the  extrados, 

0  .....  (461). 


7T> 

hypotheses  —  -=0,  p  =  r,  determine  the  minimum  of  the  pressures  P,  which 

do 

being  applied  to  a  given  point  of  the  key-stone  will  prevent  the  semi-arch  from 
turning  on  any  of  the  successive  joints  of  its  voussoirs. 


440 


THE   LINE   OF   RESISTANCE. 


When  05=0,  or  the  load  is  placed  on  the  crown  of  the 
arch, 

Y/ 1      2 
\^-a  -f 

?r~" 


Y 

l'¥ 


x 
"When  - 


/        ¥ 
(  tan.  -^  — 


cot. 


\          Y 

I  =  0,-^-  becomes  infinite  ; 


an  infinite  load  is  therefore  required  to  give  that  value  to  the 
angle   of  rupture  which  is   determined  by  this   equation. 

Y 
Solved  in  respect  to  tan.  — ,  it  gives, 

2 


tan — 

2 


.  .  (463). 


No  loading  placed  upon  the  arch  can  cause  the  angle  of  rup- 
ture to  exceed  that  determined  by  this  equation. 


THE  LINE  OF  RESISTANCE  IN  A  CIRCULAR  ARCH  WHOSE 
VOUSSOIRS  ARE  EQUAL,  AND  WHOSE  LOAD  IS  DISTRIBUTED 
OVER  DIFFERENT  POINTS  OF  ITS  EXTRADOS. 

338.  Let  it  be  supposed  that  the  pressure  of  the  load  is 
wholly  vertical,  and  such  that  any 
portion  FT  of  the  extrados  sustains 
the  weight  of  a  mass  GFTY  imme- 
diately superincumbent  to  it,  and 
bounded  by  the  straight  line  GY 
inclined  to  the  horizon  at  the  an- 
gle t;  let,  moreover,  the  weight 
of  each  cubical  unit  of  the  load  be 
equal  to  that  of  the  same  unit  of 
the  material  of  the  arch,  multiplied 
by  the  constant  factor  ^  ;  then,  re- 
presenting AD  by  K/3,  ACF  by  @, 
ACT  by  0,  and  DZ  by  z,  we  have, 


area  GFTY 


=/w. 

e 


dz: 


SEGMENTAL   ARCH. 


441 


but  TY=  MZ-(MT+  YZ),  and  MZ  =  CD  =  E-hE/3,  MT= 
E  cos.  0,  YZ  =  DZ  tan.  *  =  E  sin.  6  tan.  t.  Therefore  MT+ 
YZ  —  E  cos.  0+E  sin.  6  tan.  i  =  E  {cos.  6  cos.  *+sin.  0  sin.  1} 
sec.  t  =  E  cos.  (0—i)  sec.  *  ; 

/.  TY=  E{1  +  /3-cos.  (0-*)  sec.  1}  ; 

also,  s=DZ=E  sin.  0; 

0  0 

.-.  areaGFTY=y*TY  .  ^de=wJ*{l+(3- 

0  e 

cos.  (0—*)  sec.  *}  cos.  0d0  ; 

e 

.-.  Y=weight  of  mass  GFTY=f*Ea  C{1  +/3—  see.*  cos.(0—«)f 

e 

cos.  0^0  =  fxE3  1  (1  +)8)  (sin.  0—  sin.  0)—  i  sec.  *  {sin.  (2  B—i)  — 


sin.(20-t)}—  J(0-0)     ...  (464).* 

e 
=  moment  of  GFTY=  ^E3  /*  {(1  +13)—  sec.  t  cos.  (0— 


0 


sin.  0  cos.  0^0=fxE3{i(l+/3)(cos.  30-cos.  20)— J(cos.  '0  — 
cos.  '0)—- 1  tan.  i  (sin.  80— sin.  80)£  .  .  .  465).* 


A   SEGMENTAL   ARCH   WHOSE   EXTRADOS   IS   HORIZONTAL. 

-v  a        339.  As  the  simplest  case,  let  us  first 

1  i  ||  1 1^|  |  '  i-H  suppose  DY  horizontal,  the  material  of 
'"'  'i !  I  'J^TfTTlA  the  loading  similar  to  that  of  the  arch, 
and  the  crown  of  the  arch  at  A,  so 
that  i=0,  f*=l,  and  0=0.  Substi- 
tuting the  values  of  Y  and  Ya?  (equa- 
tions 464,  465)  which  result  from  these 
suppositions,  in  equation  (455),  solving 

that  equation  in  respect  to  _,  and  re- 


ducing, we  have,  — = 

r. 

*  See  Note  2,  at  end  of  PART  IY. — ED. 


442 


THE    GOTHIC    A  ECU. 


Assuming  -=—  =0  (see  note,  page  438.),  and  X  =  a,  and 
reducing, 

1(1  -2a)  cos.  3T  _  {(l_al  +  /3  +  l+ai_2a    cos. 


COS. 


In  the  case  in  which  the  line  of  resistance  passes  through 
the  bottom  of  the  key-stone,  so  that  X—  0,  equation  (466) 
becomes 


(l  +  cos.  Y)  cos.  T—  ^Ycot.  JY+J=0  ____  (468); 
whence  assuming  _4_—  0,  we  have 


Sill,  i 


-a     COS. 
=0.  .  .  .(469.) 


GOTHIC  AECH,  THE  EXTRADOS  OF  EACH  SEMI-ARCH  BEING- 
A  STRAIGHT  LINE  INCLINED  AT  ANY  GIVEN  ANGLE  TO  THE 
HORIZON,  AND  THE  MATERIAL  OF  THE  LOADING  DIFFERENT 
FROM  THAT  OF  THE  ARCH. 

340.  Proceeding  in  respect  to  this  general  case  of  the 
stability  of  the  circular  arch,  by  precisely  the  same  steps  as 
in  the  preceding  simpler  case,  we  obtain  from  equation 
(455), 


|cos.i--(l-|-A)co8.e}  ...(470) 


THE    GOTHIC    ARCH.  44:3 

in  which  equation  the  values  of  Y  and  Ya?  are  those  deter- 
mined by  substituting  Y  for  6  in  equations  (464)  and  (465). 

Differentiating  it  in  respect  to  Y,  assuming-^-  =0  (note, 
p.  438.),  and  X—  a,  we  obtain 

'-ia3—  K)  cos.  ©  sin.  T_(£a2+a)  sin.  Y  cos.  *— 

-(l+a)  COS.  ©  COS.  ¥{(*—  0) 


Y    ~Yx 
—  }1—  (1  +  a)  cos.  ^  cos.  ®}-^  +  —a  sin.  ^  +  {cos.  Y— 

.  j  1  d(Yx)    sin.  Y  <ZY) 
(l  +  «)  cos.  0}  I  -3  -^--^-^=0  .  .  .  (471). 

Y        YOJ 
Substituting  in  this  equation  the  values  of  -5  and  —5  ,  de 

termined  by  equations  (464)  and  (465)  the  following  equa 
tion  will  be  obtained  after  a  laborious  reduction  :  it  deter 
mines  the  value  of  Y  : 


A+B  cos.  T—  C  cos.3  Y  +  D  cos.3  T+E  sin.  Y— 
F  sin.  Y  cos.  T—  G  sin.3  Y—  H  cot. 


I(1-K  cos.  T)+a=0  ....  (472) 
where 

")  tan  f  sin8  0-(l+/3)  J2-(l+«) 

—  |(l+a)  COS.8  0  i  +(2a  +  a2—  a8—  fa4)  COS.  0 

COS.  0-(l~f*)}  +1. 

l-2a    COS.  ©. 


a)2(l—  2a)  tan.  i=fD  tan.  i. 

a)3(l  —  2a)  tan.  i  COS.  ©=E(l+a)  COS.  0. 

a)2(l—  2a)  tan.  i=~D  tan.  i. 

/3-sec.  t  cos.   ©-*     sin.  20. 


K=(l+a)  COS.  ©. 

L=M-(l  +  a)2{2(H-/3)—  sec.  i  cos.  (©—  «)}sin.  ©. 

Tables  might  readily  be  constructed  from  this  or  any  of 


44:4        AN    ARCH   SUSTAINING  THE   PRESSURE   OF   WATER. 

the  preceding  equations  by  assuming  a  series  of  values  of  ¥, 
and  calculating  the  corresponding  values  of  (3  for  each  given 
value  of  a,  *,  M-,  ®.  The  tabulated  results  of  such  a  series  of 
calculations  would  show  the  values  of  Y  corresponding  to 
given  values  of  a,  /3,  t,  (*,  ©.  These  values  of  Y  being  sub- 
stituted in  equation  (470),  the  corresponding  values  of  the 
horizontal  thrust- would, be  determined,  and  thence  the  polar 
equation  to  the  line  of  resistance  (equation  454). 


A  CIRCULAR  ARCH  HAVING  EQUAL   VOUSSOIRS  AND    SUSTAINING 
THE   PRESSURE   OF 'WATER. 

341.  Let  us  next  take  a  case  of  oblique  pressure  on  the 
extrados,  and  let  us  suppose  it  to  be 
the  pressure  of  water,  whose  surface 
stands  at  a  height  /3R  above  the  sum- 
mit of  the  key-stone.  The  pressure  of 
this  water  being  perpendicular  to  the 
extrados  will  everywhere  have  its  di- 
rection through  the  centre  C,  so  that  its 
motion  about  that  point  will  vanish, 
and  Y#— Xy=0;  moreover,  by  the 
principles  of  hydrostatics,*  the  vertical 
component  Y  of  the  pressure  of  the  water,  superincumbent 
to  the  portion  AT  of  the  extrados,  will  equal  the  weight  of 
that  mass  of  water,  and  will  be  represented  by  the  formula 
(464),  if  we  assume  i=0.  The  horizontal  component  Xf  of 
the  pressure  of  this  mass  of  water  is  represented  by  the 
formula 


-cos  .    sn.      =^    +«  +  /     cos.  ©- 

0 

cos.  0)—  J(cos.  2©—  cos.  24)J  .....  (473). 

Assuming  then  ©=0,  we  have  (equation  464),  in  respect 
to  that  portion  of  the  extrados  which  lies  between  the  crown 
and  the  points  of  rupture, 

Y 

sin.  T-    sin.  2  Y- 


and  (equation  473)  —  =^(1  +  a)9  {(1+/3)  vers.  Y—  -J  sin. 


*  See  Hydrostatics  and  Hydrodynamics,  p.  30,  31. 
|  See  Note  3,  end  of  PART  IV.—  ED. 


AN   AIICR   SUSTAINING    THE   PRESSURE   OF   WATER.         445 


...      sin.  y_      cos.  Y=Kl+a)l+/     vers.  Y_T  sn.  v 

......  (474). 

Substituting  this  value  in  equation  (455),  making  Yx—  Xy=0, 

P  #> 

solving  that  equation  in  respect  to  —  and  making  —  =1+X, 

we  have 


a_|_vers.  * 

If,  instead  of  supposing  the  pressure  of  the  water  to  be 
borne  by  the  extrados,  we  suppose  it  to 
take  effect  upon  the  intrados,  tending  to 
blow  up  the  arch,  and  if  /3  represent  the 
__^^_  __     height  of  the  water  above  the  crown  of 
the  intrados,  we  shall  obtain  precisely 
the   same  expressions  for  X  and  Y  as 
M   before,  except  that  r  must  be  substituted 
m  for  (1  +  aV,  and  X  and  Y  must  be  taken 

Y  X 

negatively  ;   in  this   case,  therefore,  -5-  sin.  Y ,-  cos.  ¥= 

— f*}(l  +  /3)  vers.  ¥—• |T  sin.  Y}  ;  whence,  by  substitution  in 
equation  (455),  and  reduction, 

P     (-ia'  +  a-f-^Ysin.Y—  ja_f-a3-f  ias-f  fA(l_f-/3)}vers.Y  , 

> .  (4:70) 


Now  by  note,  page  438,  -—^—  =0  ;   differentiating  equa- 
tions (475)  and  (476),  therefore,  and  reducing,  we  have, 

tan.     —  Xcot.^     —vers.  Y+Ax=0  .....  (477); 


which  equation  applies  to  both  the  cases  of  the  pressure  of  a 
fluid  upon  an  arch  with  equal  voussoirs  ;  that  in  which  its 
pressure  is  borne  by  the  extrados,  and  that  in  which  it  is 
borne  by  the  intrados  ;  the  constant  A  representing  in  the 


,  ,., 

first  case  the  quantity  ---  1  3~7    —  v~7T~7  —  \*  —  j  and  in  the 

20.  +a  —  tjf^l-l-a) 

K  +  X  +  KH-/3)      ™  .1       r 
second    case  —  --*—  —  .     If  the  line  of  resistance 

Ja  +«-f"t* 

pass  through  the  summit  of  the  key-stone,  X  must  be  taken  =a, 


446  EQUILIBRIUM   OF   AN   ARCH. 

If   it    pass    along    the    inferior    edge    of    the    key-stone, 

Y 

>  ^=0.     In  this  second  case,  tan.  —  {Y—  sin.  "¥}  —  0,  therefore, 

¥1=0;  so  that  the  point  of  rupture  is  at  the  crown  of  the 
arch.  For  this  value  of  Y  equations  (475)  and  (476)  become 
vanishing  fractions,  whose  values  are  determined  by  known 
methods  of  the  differential  calculus  to  be,  when  the  pressure 
is  on  the  extrados, 


<*)'....  (4T8); 
when  the  pressure  is  on  the  intrados, 

=«-f,'-ft»  .....  (479). 


It  is  evident  that  the  line  of  resistance  thus  passes  through 
the  inferior  edge  of  the  key-stone,  in  that  state  of  its  equili- 
brium which  precedes  its  rupture,  by  the  ascent  of  its  crown. 
The  corresponding  equation  to  the  line  of  resistance  is  deter- 

,  P 
mined  by  substituting  the  above  values  of  —  in    equation 

(454).     In  the  case  in  which  the  pressure  of  the  water  is 
sustained  by  the  intrados,  we  thus  obtain,  observing  that 

X  X 

—  sinJ  --  3  cos.  &=—  (*|(1  +/3)  vers.  6—^6  sin.  dj; 

_  aa  +  2a-/fy-(£a8+aa+a)cos.  6 

=r2----^         -  '-  *  '  *  (       '* 


If  for  any  value  of  6  in  this  equation,  less  than  the  angle 
of  the  semi-arch,  the  corresponding  value  of  p  exceed 
(l+ay,  the  line  of  resistance  will  intersect  the  extrados,  c 
the  arch  will  blow  up. 


THE    EQUILIBRIUM   OF   AN   ARCH,    THE   CONTACT   OF   WHOSE 
VOUSSOIRS   IS    GEOMETRICALLY   ACCURATE. 

342.   The  equations  (459)  and  (456)  completely  determine 


EQUILIBRIUM   OF   AN   ARCH. 


447 


N| 


the  value  of  P,  subject  to  the  first 
of  the  two  conditions  stated  in 
Art.  333.,  viz.  that  the  line  of  re- 
sistance passing  through  a  given 
point  in  the  key-stone,  determined 
by  a  given  value  of  \  shall  have  a 
point  of  geometrical  contact  with 
the  intrados.  It  remains  now  to 
determine  it  subject  to  the  second 
condition,  viz.  that  its  point  of  ap- 
plication P  on  the  key-stone  shall 
be  such  as  to  give  it  the  least  va- 
lue which  it  can  receive  subject  to 
the  first  condition.  It  is  evident 
that,  subject  to  this  first  condition,  every  different  value  of 
X  will  give  a  different  value  of  "V  ;  and  that  of  these  values  of 
"Y  that  which  gives  the  least  value  of  P,  and  which  corres- 
ponds to  a  positive  value  of  X  not  greater  than  a,  will  be  the 
true  angle  of  rupture,  on  the  hypothesis  of  a  mathematical 
adjustment  of  the  surfaces  of  the  voussoirs  to  one  another. 
To  determine  this  minimum  value  of  P,  in  respect  to  the  va- 
riation of  Y  dependent  on  the  variation  of  X  or  of  p,  let  it  be 
observed  that  X  does  not  enter  into  equation  (456) ;  let  that 
equation,  therefore,  be  differentiated  in  respect  to  P  and  Y, 

and  let  -=-  be  assumed =0,  and  Y  constant,  we  shall  thence 
du  * 

obtain  the  equation 


sec.  *  = 


whence,  observing  that 


sec. 


we  obtain  by  elimination  in  equation  (456) 

4T 


sin.  2Y— 2Y= 


:  —  20 


(482), 


from  which  equation  Y  may  be  determined.     Also  by  equa- 
tion (481) 


448     APPLICATIONS  OF  THE  THEORY  OF  THE  ARCH. 


----  (483); 

and  by  eliminating  sec.  Y  between  equations  (457)  and  (481), 
and  reducing, 


cos.  e=£  )  |/a(a+2)  { 


Yi»     ) 

.©--prl  >.   .   .   (484). 


The  value  of  X  given  b  y  this  equation  determines  the  actual 
direction  of  the  line  of  resistance  through  the  key-stone,  on 
the  hypothesis  made,  only  in  the  case  in  which  it  is  a  positive 
quantity,  and  not  greater  than  a  ;  if  it  be  negative,  the  line 
of  resistance  passes  through  the  bottom  of  the  key-stone,  or 
if  it  be  greater  than  a,  it  passes  through  the  top. 

Such  a  mathematical  adjustment  of  the  surfaces  of  contact 
of  the  voussoirs  as  is  supposed  in  this  article  is,  in  fact,  sup- 
plied by  the  cement  of  an  arch.  It  may  therefore  be  con- 
sidered to  involve  the  theory  of  the  cemented  arch,  the  influ- 
ence on  the  conditions  of  its  stability  of  the  adhesion  of  its 
voussoirs  to  one  another  being  neglected.  In  this  settlement, 
an  arch  is  liable  to  disruption  in  some  of  those  directions  in 
which  this  adhesion  might  be  necessary  to  its  stability.  That 
old  principle,  then,  which  assigns  to  it  such  proportions  as 
would  cause  it  to  stand  firmly  did  no  such  adhesion  exist, 
will  always  retain  its  authority  with  the  judicious  engineer. 


APPLICATIONS  OF  THE  THEORY  OF  THE  ARCH. 

343.  It  will  be  observed  that  equation  (459)  or  (472) 
determines  the  angle  "f  of  rupture  in  terms  of  the  load  Y, 
and  the  horizontal  distance  x  of  its  centre  of  gravity  from 
the  centre  C  of  the  arch,  its  radius  r,  and  the  depth  &r  of  its 
voussoirs  ;  moreover,  that  this  determination  is  wholly  inde- 
pendent of  the  angle  of  the  arch,  and  is  the  same  whether 
its  arc  be  the  half  or  the  third  of  a  circle ;  also,  that  if  the 
angle  of  the  semi-arch  be  less  than  that  given  by  the  above 
equation  as  the  value  of  Y,  there  are  no  points  of  rupture, 
such  as  they  have  been  defined,  the  line  of  resistance  passing 
through  the  springing  of  the  arch  and  cutting  the  iiitradoa 
there. 


THEORY   OF  THE  ARCH.  449 

The  value  of  Y  being  known  from  this  equation,  P  is 
determined  from  equation  (456),  and  this  value  of  P  being 
substituted  in  equation  (454),  tne  line  of  resistance  is  com- 
pletely determined ;  and  assigning  to  d  the  value  ACB 
(p.  437.),  the  corresponding  value  of  p  gives  us  the  position 
of  the  point  Q,  where  the  line  of  resistance  intersects  the 
lowest  voussoir  of  the  arch,  or  the  summit  of  the  pier. 
Moreover,  P  is  evidently  equal  to  the  horizontal  thrust  on 
the  top  of  the  pier,  and  the  vertical  pressure  upon  it  is  the 
weight  of  the  arch  and  load:  thus  all  the  elements  are 
known,  which  determine  the  conditions  of  the  stability  of  a 
pier  or  buttress  (Arts.  293.  and  312.)  of  given  dimensions 
sustaining  the  proposed  arch  and  its  loading. 

Every  element  of  the  theory  of  the  arch  and  its  abutments. 
is  involved,  ultimately,  in  the  solution  in  respect  to-  V  of 
equation  (459)  or  equation  (472).  Unfortunately  this  solu- 
tion presents  great  analytical  difficulties..  In  the-  failure  of 
any  direct  means  of  solution,  there  are,  however,  various 
methods  by  which  the  numerical1  relation  of  Y  and  Y  may 
be  arrived  at  indirectly.  Among  thei%  one  of  the  simplest 
is  this : — 

Let  it  be  observed  that  that  equation  is  readily  soluble  in 
respect  to  Y ;  instead,  then,  of  determining  the  value  of  Y 
for  an  assumed  value  of  Y,  determine  conversely  the  value 
of  Y  for  a  series  of  assumed  values  of  Y.  Knowing  the  dis- 
tribution of  the  load  Y,  the  values  of  as  will  be  known  in 
respect  to  these  values  of  Y,  and  thus  the  values  of  Y  may 
be  numerically  determined,  and  may  be  tabulated.  From 
such  tables  may  be  found,  by  inspection,  values  of  Y  corres- 
ponding to  given  values  of  i . 

The  values  of  Y,  P,  and  r  are  completely  determined  by 
equations  (482,  483,  484),  and  all  the  circumstances  of  the 
equilibrium  of  the  circular  arch  are  thence  known,  on  the 
hypothesis,  there  made,  of  a  true  mathematical  adjustment 
of  the  surfaces  of  the  voussoirs  to  one  another ;  and  although 
this  adjustment  can  have  no  existence  in  practice  when, 
the  voussoirs  are  put  together  without  cement,  yet  may  it 
obtain  in  the  cemented  arch.  The  cement,  by  reason  of 
its  yielding  qualities  when  fresh,  is  made  to  enter  into  so 
intimate  a  contact  with  the  surfaces  of  the  stones  between 
which  it  is  interposed  that  it  takes,  when  dry,  in  respect 
to  each  joint  (abstraction  being  made  of  its  adhesive  proper- 
ties), the  character  of  an  exceedingly  thin  voussoir,  having 
its  surfaces  mathematically  adjusted  to  those  of  the  adjacent 
v^ussoirs ;  so  that  if  we  imagine,  not  the  adhesive  properties 

29 


450  APPLICATIONS   OF  THE 

of  the  cement  of  an  arch,  but  only  those  which  tend  to  tho 
more  uniform  diffusion  of  the  pressures  through  its  mass,  to 
enter  into  the  conditions  of  its  equilibrium,  these  equations 
embrace  the  entire  theory  of  the  cemented  arch.  The  hypo- 
thesis here  made  probably  includes  all  that  can  be  relied 
upon  in  the  properties  of  cement  as  applied  to  large  struc- 
tures. 

An  arch  may  FALL  either  by  the  sinking  or  the  rising  of 
its  crown.  In  the  former  case,  the  line  of  resistance  passing 
through  the  top  of  the  key-stone  is  made  to  cut  the  extrados 
beneath  the  points  of  rupture ;  in  the  latter,  passing  through 
the  bottom  of  the  key-stone,  it  is  made  to  cut  the  extrados 
between  the  points  of  rupture  and  the  crown. 

In  the  first  case  the  values  of  X,  Y,  and  P,  being  deter- 
mined as  before  and  substituted  in  equation  (454),  and  p 
being  assumed  =  (!+«)?•,  the  value  of  d,  which  corresponds 
to  p=(l-f-a)r,  will  indicate  the  point  at  which  the  line  of 
resistance  cuts  the  extrados.  If  this  value  of  d  be  less  than 
the  angle  of  the  semi-arch,  the  intersection  of  the  line  of 
resistance  with  the  extrados  will  take  place  above  the 
springing,  and  the  arch  will  fall. 

In  the  second  case,  in  which  the  crown  ascends,  let  the 
maximum  value  of  p  be  determined  from  equation  (454),  p 
being  assumed  =r ;  if  this  value  of  p  be  greater  than  R,  and 
the  corresponding  value  of  &  less  than  the  angle  of  rupture, 
the  line  of  resistance  will  cut  the  extrados,  the  arch  will 
open  at  the  intrados,  and  it  will  fall  by  the  descent  of  the 
crown. 

If  the  load  be  collected  over  a  single  point  of  the  arch, 
the  intersection  of  the  line  of  resistance  with  the  extrados 
will  take  place  between  this  point  and  the  crown ;  it  is  that 
portion  only  of  the  line  of  resistance  which  lies  between  these 
points  which  enters  therefore  into  the  discussion.  Now  if 
we  refer  to  Art.  336.,  it  will  be  apparent  that  in  respect  to 
this  portion  of  the  line,  the  values  of  X  and  Y  in  equations 
(453)  and  (454)  are  to  be  neglected ;  the  only  influence  of 
these  quantities  being  found  in  the  value  of  P. 


THEOKY   OF   THE   ARCH. 


451 


Example  1. — Let  a  circular  arch  of  equal  voussoirs  have 
the  depth  of  each  voussoir  equal  to 

ITVth  the  diameter  of  its  intrados,  so 
r  that  a=*2,  and  let  the  load  rest  upon 
_^     it   by  three   points  A,  B,  D  of  its 

JA  extrados,  of  which  A  is  at  the  crown 
and  B  D  are  each  distant  from  it  45° ; 
and  let  it  be  so  distributed  that  -fths 
of  it  may  rest  upon  each  of  the  points 
B  and  D,  and  the  remaining  J  upon 
A  ;  or  let  it  be  so  distributed  within 
60°  on  either  side  of  the  crown  as  to 
produce  the  same  effect  as  though  it 
rested  upon  these  points. 

Then  assigning  one  half  of  the  load 
upon  the  crown  to  each  semi-arch, 
and  calling  x  the  horizontal  distance 
of  the  centre  of  gravity  of  the  load  upon  either  semi-arch 

from  C,  it  may  easily  be  calculated  that  -  =  £  sin.  45  = 

•5303301.  Hence  it  appears  from  equation  (463)  that  no 
loading  can  cause  the  angle  of  rupture  to  exceed  65°. 
Assume  it  to  equal  60°;  the  amount  of  the  load  necessary  to 
produce  this  angle  of  rupture,  when  distributed  as  above, 
will  then  be  determined  by  assuming  in  equation  (460), 

^=60°,  and  substituting  a  for  X,  -2  for  a,  and  -5303301  for?. 

Y  Y 

"We  thus  obtain  -^=-0138.    Substituting  this  value  of  -,,  and 

also  the  given  values  of  a  and  T  in  equation  (457),  and 

observing  that  in  this  equation  £-  is  to  be  taken  =1+ a  and 

r 

P  P 

0=0,  we  find -5  =  -11832.     Substituting  this  value  of  —  in 

the  equation  (454),  we  have  for  the  final  equation  to  the  line 
of  resistance  beneath  the  point  B 


•2426  vers.^H- -1493 


•0138  sin.  &  +  -1183  cos.  &  +  -22  &  sin.  0* 


452 


APPLICATIONS   OF   THE 


If  the  arc  of  the -arch  be  a  com- 
plete semicircle,  the  value  of  p  in  this 

equation  corresponding  to  d  =  -  will 

a 

determine  the  point  Q,  where  the 
line  of  resistance  intersects  the  abut- 
ment; this  value  is  p=~L'Q9r. 

If  the  arc  of  the  arch  be  the  third 
o  of  a  circle,  the  value  of  p  at  the 
abutment   is  that  corresponding  to 

&  =  - ;  this  will  be  found  to  be  r,  as 
o 

it  manifestly  ought  to  be,  since  the 
points  of  rupture  are  in  this  case  at 
the  springing. 

In  the  first  case  the  volume  of  the  semi-arch  and  load  is 
represented  by  the  formula 


1  ,    ' 

l^[ 

/I 

!/ 

I 

/  1 

;1 

i 

/     1 

•J- 

/I 

r     1 

1 

! 

1 

and  in  the  second  case  by 


Thus,  supposing  the  pier  to  be  of  the  same  material  as  the 
arch,  the  volume  of  its  material,  which  would  have  a  weight 
equal  to  the  vertical  pressure  upon  its  summit,  would  in  the 
first  case  be  *3594r2,  and  in  the  second  case  -2442^2,  whilst 
the  horizontal  pressures  P  would  in  both  cases  be  the  same, 
viz.  -11832?'2  ;  substituting  these  values  of  the  vertical  and 
horizontal  pressures  on  the  summit  of  the  pier,  in  equation 
(377),  and  for  Ic  writing  -J-  a—  (p—  r\  we  have  in  the  first 
case 


n[_ 


_  -3594(0— 


and  in  the  second  case, 
H= 


-2442  ar* 


THEORY   OF   THE   AECH.  453 

where  H  is  the  greatest  height  to  which  a  pier,  whose  width 
is  a,  can  be  built  so  as  to  support  the  arch. 

If  \tf— -11832^=0,  or  «=-4864r,  then  in  either  case  the 
pier  may  be  built  to  any  height  whatever,  without  being 
overthrown.  In  this  case  the  breadth  of  the  pier  will  be 
nearly  equal  to  Jth  of  the  span. 

The  height  of  the  pier  being  given  (as  is  commonly  the 
case),  its  breadth,  so  that  the  arch  may  just  stand  firmly 
upon  it,  may  readily  be  determined.  As  an  example,  let  us 
suppose  the  height  of  the  pier  to  equal  the  radius  of  the 
arch.  Solving  the  above  equations  in  respect  to  a,  we  shall 
then  obtain  in  the  first  case  en  =  'SOTS/1,  and  in  the  second 
a='3r. 

If  the  span  of  each  arch  be  the  same,  and  rl  and  r2  repre- 
sent their  radii  respectively,  then  rl=r9  sin.  60*  ;  supposing 
then  the  height  of  the  pier  in  the  second  arch  to  be  the  same 
as  that  in  the  first,  viz.  r^  then  in  the  second  equation  we 
must  write  for  H,  r^  sin.  60°.  We  shall  thus  obtain  for  a  the 
value  "28/v 

The  piers  shown  by  the  dark  lines  in  the  preceding 
figures  are  of  such  dimensions  as  just  to  be  sufficient  to 
sustain  the  arches  which  rest  upon  them,  and  their  loads, 
both  being  of  a  height  equal  to  the  radius  of  the  semicircular 
arch.  It  will  be  observed,  that  in  both  cases  the  load 
Y='0138r2,  being  that  which  corresponds  to  the  supposed 
angle  of  rupture  60°,  is  exceedingly  small. 


Example  2.— Let  us  next  take  the  example  of  a  Gothic 
arch,  and  let  us  suppose,  as  in  the  last  examples,  that  the 
angle  of  rupture  is  60°,  and  that  a='2;  but  let  the  load  in 
this  case  be  imagined  to  be  collected  wholly  over  the 

crown  of  the  arch,  so  that  -  =  sin.  30°.   Substituting  in  equa- 
tion (459),  30°  for  0,  and  60°  for  T,  and  -2  for  a,  and  sin.  30° 

for  -,  we  shall  obtain  the  value  *21015  for  — - ;  whence  bv 

T  r* '  J 

p 

equation  (457)-^-  =  -2405,  and  this  value  being  substituted, 


454 


APPLICATIONS    OF   THE 


equation  (454)  gives  1*1457'  foi 

the  value  of  p  when  0  =  _.    "We 

2 

have  thus  all  the  data  for  deter- 
mining the  width  of  a  pier  of 
given  height  which  will  just 
support  the  arch.  Let  the 
height  of  the  pier  be  supposed, 
as  before,  to  equal  the  radius 
of  the  intrados  ;  then,  since  the 
weight  of  the  semi-arch  and  its 
load  is  •5556r2,  and  the  horizon- 
tal thrust  -2405712,  the  width  a 
of  the  pier  is  found  by  equation 
(379)  to  be  «4195r. 
The  preceding  figure  represents  this  arch ;  the  square, 
formed  by  dotted  lines  over  the  crown,  shows  the  dimensions 
of  the  load  of  the  same  materials  as  the  arch  which  will  cause 
the  angle  of  the  rupture  to  become  60°  ;  the  piers  are  of  the 
required  width  ^l&S/1,  such  that  when  their  height  is  equal 
to  AB,  as  shown  in  the  figure,  and  the  arch  bears  this  insist- 
ent pressure,  they  may  be  on  the  point  of  overturning. 


TABLES  OF  THE  THRUST  OF  ARCHES. 

344.  It  is  not  possible,  within  the  limits  necessarily 
assigned  to  a  work  like  this,  to  enter  further  upon  the  dis- 
cussion of  those  questions  whose  solution  is  involved  in  the 
equations  which  have  been  given  ;  these  can,  after  all,  be- 
come accessible  to  the  general  reader,  only  when  tables  shall 
be  formed  from  them. 

Such  tables  have  been  calculated  with  great  accuracy  by 
M.  Garidel  in  respect  to  that  case  of  a  segmental  arch*  whose 
loading  is  of  the  same  material  as  the  voussoirs,  and  the  ex- 
trados  of  each  semi-arch  a  straight  line  inclined  at  any  given 
angle  to  the  horizon.  These  tables  are  printed  in  the  Ap- 
pendix (Tables  2,  3). 


*  The  term  segmental  arch  is  used,  here  and  elsewhere,  to  distinguish  that 
form^of  the  circular  arch  in  which  the  intrados  is  a  contiguous  segment  from 
that  in  which  it  is  composed  of  two  segments  struck  from  different  centres,  aa 
in  the  Gothic  arch. 


THEORY   OF  THE   ARCH.  455 

Adopting  the  theory  of  Coulomb*,  M.  Garidel  has  arrived 
at  an  equationf  which  becomes  identical  with  equation  (472) 
in  respect  to  that  particular  case  of  the  more  general  condi- 
tions embraced  by  that  equation,  in  which  ^=1  and  0=0. 

By  an  ingenious  method  of  approximation,  for  the  details 
of  which  the  reader  is  referred  to  his  work,  M.  Garidel  has 
determined  the  values  of  the  angle  of  rupture  Y,  and  the 

p 
quantity  — ,  in  respect  to  a  series  of  different  values  of  a  and 

(3.  The  results  are  contained  in  the  tables  which  will  be 
found  at  the  end  of  this  volume. 

p 
The  value  of  -^  being  known  from  the  tables,  and  the 

values  of  Y  and  Ya?  from  eouations  (464),  (465),  the  line  of 
resistance  is  determined  by  the  substitution  of  these  values 
in  equation  (454).  The  line  of  resistance  determines  the 
point  of  intersection  of  the  resultant  pressure  with  the  sum- 
mit of  pier  ;  the  vertical  and  horizontal  components  of  this 
resultant  pressure  are  moreover  known,  the  former  being  the 
weight  of  the  semi-arch,  and  the  other  the  horizontal  thrust 
on  the  key.  All  the  elements  necessary  to  the  determina- 
tion of  the  stability  of  the  piers  (Arts.  289  and  312)  are 
therefore  known. 

It  will  be  observed  that  the  amount  of  the  horizontal 
thrust  for  each  foot  of  the  width  of  the  soffit  is  determined 

p 
by  multiplying  the  value  of  — a,  shown  by  the  tables,  by  the 

square  of  the  radius  of  the  intrados  in  feet,  and  by  the 
weight  of  a  cubic  foot  of  the  material 


*  See  Mr.  Hann's  Theory  of  Bridges,  Art.  16. ;  also  p.  24.  of  the  Memoir  on 
the  Arch  by  the  author  of  this  work,  contained  in  the  same  volume, 
f  Tables  des  Poussees  des  Voutes,  p.  44.    Paris,  1837.    Bachelier. 


456  NOTES   TO  PAET  IV. 


NOTE  1.    PART  IY. 

The  length  of  an  elementary  arc  ds  of  the  intrados  AS  subtending  the  angle 
dd  is  expressed  by  rdd ;  an  elementary  volume  of  the  arch  will  therefore  be 
expressed  by  rdftdr ;  the  perpendicular  distance  of  the  centre  of  gravity  of 
this  volume  from  the  vertical  line  CE  is  r  sin.  0 ;  the  moment  of  this  volume, 
with  regard  to  CE,  is  therefore  rdddr^r  sin.0=rVr  sin.  6d6;  then  from  (Art. 
31.)  equation  (20)  there  obtains 

R         0 
Wi=fr*drjlm.  Odd. 


NOTE  2.     PART  IV.  —  General  integrals  of  equations  464,  465. 
^Phe  general  integral,  (equation  464) 

y{l-|_£_co8.  (0-fr)  sec.  i  }  cos.  6dd=f(l+(3)  cos.  6dff— 

/sec.  i  (cos.  0  cos.  i+sin.  6  sin.  i)  cos.  W0=  /(l-f-0)  cos.  Odd— 

j  sec.  i  cos.  i  cos.20oR9—  /  see.  i  sin.  i  sin.  6  cos.  0d0. 
But^y  (l-f-/3)  cos.  0o?0=(l-f-/?)  sin.  0;    /sec.  t  cos.  t  cos.  *6dd= 

.  t  cos.  i^y  (-   |  °OS2    )c?0=sec.  t  cos.t(-0-f-  sin.  20);  /sec.  t  sin  t 

/j  j 

-sin.  20cf(20)  =  —  sec.  i  sin.  t  —  cos.  20; 
4  4 

,-.J    |  1+/3-COS.  (0-t)  sec.  i  j-cos.  0c?0=(l-j-/?)  sin.  0-i  sec.  z 
(sin.  20  cos.  t—  sin.t  cos.  20)-^0=:(l+)8)  sin.  0-^sec.  tsin.  (20-4)-^ 

The  general  integral,    /  {(1  -}-/?)—  sec.  t  cos.  (0—  t)}  sin.  0  cos.  0<f0,  (equation 
465),  =y(l  -f/3)  sin.  0  cos.  Odd—  i*  sec.  i  {cos.Pcos.  f-f-sin.08in.  i}sin.0  cos.0^0. 


sec 


cos.  20= 


NOTES  TO   PART  IV.  457 


/%ec.  tj  cos.  6  cos.  t-fsin.  0sin.  1 j  sin.  6  cos.  fid .  0==y  co8.20sin. 
/tan.  i  sin.  U0cos.  6dd  .  — /cos.20dcos.  0-f-/tan.  *  sin-  2^  sin-  0= 


_LCOS.  "0_f_I  sin.  "0 
3  o 


...  f  |  (14.5)  -see.  i  cos.  (0-0  j-  sin.  0  cos.  0rf  .  -|-(f0=  -^(1-f/?)  cos.  3 
L  (l-|-/3)4-  J-  cos.  30-i-  tan.  t  sin.  80 .  tan  t. 

4  o  o 


NOTE  3.    PART  IV. 

In  equation  (427),  (Art.  319),  by  making  0=0,  we  obtain  P=£  Hi  x* ;  since 
tan.  r=l,  and  this  answers  to  the  case  of  the  horizontal  pressure  of  a  perfect 

fluid  like  water.  From  this  expression  there  obtains  dl*=/j.ixdxj  to  express 
the  elementary  pressure  at  any  depth  x  below  the  surface.  This  depth  in 
(Art.  341),  equation  (473),  is  TV=AD+AB=AD-f  AC-BC=/3R-f  R-Rcos.0, 


.-.  dP=iiR(l4-{3— cos.  6)Rd (1-f-/?— cos.  0)=//R2  J1+/3— cos.  0f  sin. 

*  *  ) 

0 
.-.  P=X=yuR2>/>  Jl-f /3-cos.  0  }  sin.  0d0. 

e 


T   V. 

THE   STRENGTH  OF  MATERIALS. 


ELASTICITY. 

345.  From  numerous  experiments  which  have  been  made 
upon  the  elongation,  flexure,  and  torsion  of  solid  bodies 
under  the  action  of  given  pressures,  it  appears  that  the 
displacement  of  their  particles  is  subject  to  the  following 
laws. 

1st.  That  when  this  displacement  does  not  extend  beyond 
a  certain  distance,  each  particle  tends  to  return  to  the  place 
which  it  before  occupied  in  the  mass,  with  a  force  exactly 
proportional  to  the  distance  through  which  it  has  been 
displaced. 

2dly.  That  if  this  displacement  be  carried  beyond  a 
certain  distance,  the  particle  remains  passively  in  the  new 
position  which  it  has  been  made  to  take  up,  or  passes  finally 
into  some  other  position  different  from  that  from  which  it 
was  originally  moved. 

The  effect  of  the  first  of  these  laws,  when  exhibited  in 
the  joint  tendency  of  the  particles  which  compose  any  finite 
mass  to  return  to  any  position  in  respect  to  the  rest  of  the 
mass,  or  in  respect  to  one  another,  from  which  they  have 
been  displaced,  is  called  elasticity.  There  is  every  reason  to 
believe  that  it  exists  in  all  bodies  within  the  limits,  more  or 
less  extensive,  which  are  imposed  by  the  second  law  stated 
above. 

The  force  with  which  each  separate  particle  of  a  body 
tends  to  return  to  the  position  from  which  it  has  been 
displaced  varying  as  the  displacement,  it  follows  that  the 
force  with  which  any  aggregation  of  such  particles,  consti- 
tuting a  finite  portion  of  the  body,  when  extended  or 
compressed  within  the  limits  of  elasticity,  tends  to  recover 
its  form,  that  is  the  force  necessary  to  keep  it  extended  or 


ELONGATION.  459 

compressed,  is  proportional  to  the  amount  of  the  extension 
or  compression ;  so  that  each  equal  increment  of  the  extend- 
ing or  compressing  force  produces  an  equal  increment  of  its 
extension  or  compression.  This  law,  which  constitutes 
perfect  elasticity,  and  which  obtains  in  respect  to  fluid  and 
gaseous  bodies  as  well  as  solids,  appears  first  to  have  been 
established  by  the  direct  experiments  of  S.  Gravesande  on 
the  elongation  of  thin  wires.* 

It  is,  however,  by  its  influence  on  the  conditions  of 
deflexion  and  torsion  that  it  is  most  easily  recognized  as 
characterizing  the  elasticity  of  matter,  under  all  its  solid 
forms, f  within  certain  limits  of  the  displacement  of  its 
particles  or  elements,  called  its  elastic  limits. 


ELONGATION. 

346.  To  determine  tTie  elongation  or  compression  of  a  bar  of 
a  given  section  under  a  given  strain. 

Let  K  be  taken  to  represent  the  section  of  the  bar  in 
square  inches,  L  its  length  in  feet,  I  its  elongation  or  com- 
pression in  feet  under  a  strain  of  P  pounds,  and  E  the  strain 
or  thrust  in  pounds  which  would  be  required  to  extend  a 
bar  of  the  same  material  to  double  its  length,  or  to  compress 

*  For  a  description  of  the  apparatus  of  S.  Gravesande,  see  Illustrations  of 
Mechanics,  by  the  Author  of  this  work,  2d  edition,  p.  30.  In  one  of  his 
experiments,  Mr.  Barlow  subjected  a  bar  of  wrought  iron,  one  square  inch  in 
section,  to  strains  increasing  successively  from  four  to  nine  tons,  and  found  the 
elongations  corresponding  to  the  successive  additional  strains,  each  of  one  ton, 
to  be,  in  millionths  of  the  whole  length  of  the  bar,  120,  110,  120,  120,  120. 
In  a  second  experiment,  made  with  a  bar  two  square  inches  in  section,  under 
strains  increasing  from  10  tons  to  30  tons,  he  found  the  additional  elongations, 
produced  by  successive  additional  strains,  each  of  two  tons,  to  be,  in  millionths 
of  the  whole  length,  110,  110,  110,  110,  100,  100,  100,  100,  95,  90.  From  an 
extensive  series  of  similar  results,  obtained  from  iron  of  different  qualities,  he 
deduced  the  conclusion  that  a  bar  of  iron  of  mean  quality  might  be  assumed 
to  elongate  by  100  millionth  parts,  or  the  10,000th  part,  of  its  whole  length, 
under  every  additional  ton  strain  per  square  inch  of  its  section.  (Report  to 
Directors  of  London  and  Birmingham  Railway.  Fellowes,  1835.) 

The  French  engineers  of  the  Pont  des  Invalides  assigned  82  millionth  parts 
to  this  elongation,  their  experiments  having  probably  been  made  upon  iron  of 
inferior  quality.  M.  Vicat  has  assigned  91  millionth  parts  to  the  elongation 
of  cables  of  iron  wire  (No.  18.)  under  the  same  circumstances,  MM.  Minard 
and  Desormes,  1,176  millionth  parts  to  the  elongation  of  bars  of  oak. 
(lllust.  Mech.,  p.  393.) 

f  The  experiments  of  Prof.  Robison  on  torsion  show  the  existence  of  this 
property  in  substances  where  it  might  little  be  expected;  in  pipe-clay,  for 
instance. 


460  THE   WOEK   EXPENDED   ON   ELONGATION. 

it  to  one  half  its  length,  if  the  elastic  limit  of  the  material 
were  such  as  to  allow  it  to  be  so  far  elongated  or  compressed. 
the  law  of  elasticity  remaining  the  same.* 

Now,  suppose  the  bar,  whose  section  is  K  square  inches, 
to  be  made  up  of  others  of  the  same  length  L,  each  one  inch 
in  section  ;  these  will  evidently  be  K  in  number,  and  th'3 

p 
strain  or  the  thrust  upon  each  will  be  represented  by  ^. 

Moreover,  each  bar  will  be  elongated  or  compressed,  by  this 
strain  or  thrust,  by  I  feet  ;  so  that  each  foot  of  the  length  of 
it  (being  elongated  or  compressed  by  the  same  quantity  as 
each  other  foot  of  its  length)  will  be  elongated  or  compressed 

by  a  quantity  represented,  in  feet,  by  y.     But  to  elongate 

or  compress  a  foot  of  the  length  of  one  of  these  bars,  by  one 
foot,  requires  (by  supposition)  E  pounds  strain  or  thrust  ;  to 

elongate  or  compress  it  by    -  feet  requires,  therefore, 


j 


pounds.     But  the  strain  or  thrust  which  actually  produces 

P  P         I 

this  elongation  is  =^  pounds.     Therefore,^  =  E^-. 

PT 


347.  To  find  the  number  of  units  of  work  expended  upon  the 
elongation  ly  a  given  quantity  (I)  of  a  bar  whose  section  is 
K  and  its  length  L. 

If  x  represent  any  elongation  of  the  bar  (x  being  a  part 
of  l\  then  is  the  strain  P  corresponding  to  that  elongation 

KE 

represented   (equation  485)  by  -y-#;   therefore   the  work 

done  in  elongating  the  bar  through  the   small  additional 

KE 
space  A#,  is  represented  by  -y-a?A#  (considering  the   strain 

to  remain  the  same  through  the  small  space  Aa?)  ;  and  the 


*  The  value  of  E  in  respect  to  any  material  is  called  the  modulus  of  its  elas- 
ticity. The  value  of  the  moduli  of  elasticity  of  the  principal  materials  of  con- 
struction  have  been  determined  by  experiment,  and  will  be  found  in  a  table  at 
the  end  of  the  volume. 


THE   WORK   EXPENDED   ON   ELONGATION.  461 

whole  work  U  done  is,  on  this  supposition,  represented  by 
-=— 2&A&,   or    (supposing    A#J    to    be    infinitely  small)    by 

KE  /   .  KE7, 

-f-J  ®d%  or  by  ITT^ 

(486). 


TTT? 

348.  By  equation   (485)  P=-pZ,    therefore    U  = 

whence  it  follows  that  the  work  of  elongating  the  bar  is  one 
half  that  which  would  have  been  required  to  elongate  it  by 
the  same  quantity,  if  the  resistance  opposed  to  its  elongation 
had  been,  throughout,  the  same  as  its  e'xtreme  elongation  I. 
If,  therefore,  the  whole  strain  P  corresponding  to  the 
elongation  I  had  been  put  on  at  once,  then,  when  the  elonga- 
tion I  had  been  attained,  twice  as  much  work  would  have 
been  done  upon  the  bar  as  had  been  expended  upon  its 
elasticity.  This  work  would  therefore  have  been  accumu- 
lated in  the  bar,  and  in  the  body  producing  the  strain  under 
which  it  yields ;  and  if  both  had  been  free  to  move  on  (as, 
for  instance,  when  the  strain  of  the  bar  is  produced  by  a 
weight  suspended  freely  from  its  extremity),  then  would 
this  accumulated  work  have  been  just  sufficient  yet  further 
to  elongate  the  bar  by  the  same  distance  Z,*  which  whole 
elongation  of  2Z  coulcl  not  have  remained;  because  the 
strain  upon  the  bar  is  only  that  necessary  to  keep  it 
elongated  by  I.  The  extremity  of  the  bar  would  therefore, 
under  these  circumstances,  have  oscillated  on  either  side  of 
that  point  which  corresponds  to  the  elongation  I. 

• 

*  The  mechanical  principle  involved  in  this  result  has  numerous  applica- 
tions ;  one  of  these  is  to  the  effect  of  a  sudden  variation  of  the  pressure  on  a 
mercurial  column.  The  pressure  of  such  a  column  varying  directly  with  its 
elevation  or  depression,  follows  the  same  law  as  the  elasticity  of  a  bar; 
whence  it  follows  that  if  any  pressure  be  thrown  at  once  or  instantaneously 
upon  the  surface  of  the  mercury,  the  variation  of  the  height  of  the  column 
will  be  twice  that  which  it  would  receive  from  an  equal  pressure  gradually 
accumulated.  Some  singular  errors  appear  to  have  resulted  from  a  neglect  of 
this  principle  in  the  discussion  of  experiments  upon  the  pressure  of  steam, 
made  with  the  mercurial  column.  No  such  pressure  can  of  course  be  made  to 
operate,  in  the  mathematical  sense  of  the  term,  instantaneously ;  and  the  term 
gradually  has  a  relative  meaning.  All  that  is  meant  is,  that  a  certain  relation 
must  obtain  between  the  rate  of  the  increase  of  the  pressure  and  the  amplitude 
of  the  motion,  so  that  when  the  pressure  no  longer  increases  the  motion  may 
cease. 


462  RESILIENCE   AND   FRAGLITY. 

349.  Eliminating  I  between  equations  (485)  and  (486),  we 
obtain 

U=ig (487); 

whence  it  appears  that  the  work  expended  upon  the  elonga- 
tion of  a  bar  under  any  strain  varies  directly  as  the  square 
of  the  strain  and  the  length  of  the  bar,  and  inversely  as  the 
area  of  its  section.* 


THE  MODULI  OF  RESILIENCE  AND  FRAGILITY. 

(7  v  a 
y  1  KL  (equation  486),  it  is  evident 

that  the  different  amounts  of  work  which  must  be  done  upon 
different  bars  of  the  same  material  to  elongate  them  by  equal 

fractional  parts  I  y),  are  to  one  another  as  the  product  KL. 


Let  now  two  such  bars  be  supposed  to  have  sustained  that 
fractional  elongation  which  corresponds  to  their  elastic  limit; 
let  U«  represent  the  work  which  must  have  been  done  upon 
the  one  to  bring  it  to  this  elongation,  and  Ma  that  upon  the 
other  :  and  let  the  section  of  the  latter  bar  be  one  square 
inch  and  its  length  one  foot  ;  then  evidently 

Ue=M,KL  .....  (488). 

M«  is  in  this  case  called  the  modulus  of  longitudinal  resili- 
ence.^ 

It  is  evidently  a  measure  of  that  resistance  which  the 
material  of  the  bar  opposes  to  a  strain  in  the  nature  of  an 
impact,  tending  to  elongate  it  beyond  its  elastic  limits. 

If  M/be  taken  to  represent  the  work  which  must  be  simi- 
larly done  upon  a  bar  one  foot  long  and  one  square  inch,  in 
section  to  produce  fracture,  it  will  be  a  measure  of  that 
resistance  which  the  bar  opposes  to  fracture  under  the  like 
circumstances,  and  which  resistance  is  opposed  to  its  fra- 

*  From  this  formula  may  be  determined  the  amount  of  work  expended  pre- 
judicially upon  the  elasticity  of  rods  used  for  transmitting  work  in  machinery, 
under  a  reciprocating  motion  —  pump  rods,  for  instance.  A  midden  effort  of  the 
pressure  transmitted  in  the  nature  of  an  impact  may  make  the  loss  of  work 
double  that  represented  by  the  formula  ;  the  one  limit  being  the  minimum,  and 
the  other  the  maximum,  of  the  possible  loss. 

f  The  term  "modulus  of  resilience"  appears  first  to  have  been  used  by 
Mr.  Tredgold  in  his  work  on  "  the  Strength  of  Cast  Iron,"  Art.  304. 


A   BAR    SUSPENDED   VERTICALLY.  4:63 

gility  ;  it  may  therefore  be  distinguished  from  the  last  men- 
tioned as  the  modulus  of  fragility.  If  TJ/  represent,  the  work 
which  must  be  done  upon  a  bar  whose  section  is  K  square 
inches  and  its  length  L  fee*  to  produce  fracture;  then,  as 
before, 

U/=M/KL  .....  (489). 

If  Pe  and  P/  represent  respectively  the  strains  which 
would  elongate  a  bar,  whose  length  is  L  feet  and  section^  K 
inches,  to  its  elastic  limits  and  to  rupture  ;  then,  equation 
(487), 


"M<=1  Similarly  Mr=i          .....  (490). 

These  equations   serve  to  determine  the  values  of  the 
moduli  Me  and  M/by  experiment.* 


351.  The  elongation  of  a  lor  suspended^  vertically,  and  sus- 
taining a  given  strain  in  the  direction  of  its  length,  the 
influence  of  its  own  weight  being  taken  into  the  account. 

Let  x  represent  any  length  of  the  bar  before  its  elonga- 
tion, &x  an  element  of  that  length,  L  the  whole  length  of  the 
bar  before  elongation,  w  the  weight  of  each  foot  of  its 
length,  and  K  its  section.  Also  let  the  length  x  have  become 
a?x  when  the  bar  is  elongated,  under  the  strain  P  and  its  own 
weight.  The  length  of  the  bar,  below  the  point  whose  dis- 
tance from  the  point  of  suspension  was  x  before  the  elonga- 
tion, having  then  been  L— a?,  and  the  weight  of  that  portion 
of  the  bar  remaining  unchanged  by  its  elongation,  it  is  still 
represented  by  (L— a?)  w.  Now  this  weight,  increased  by  P, 
constitutes  the  strain  upon  the  element  AOJ;  its  elongation 
under  this  strain  is  therefore  represented  (equation  485)  by 

—     -rr-n     — A#>  and  the  length  ^xl  of  the  element  when  thus 

*  The  experiments  required  to  this  determination,  in  respect  to  the  princi- 
pal materials  of  construction,  have  been  made,  and  are  to  be  found  in  the 
published  papers  of  Mr.  Hodgkinson  and  Mr.  Barlow.  A  table  of  the  moduli 
of  resilience  and  fragility,  collected  from  these  valuable  data,  is  a  desideratunc 
in  practical  science. 


464  THE   VERTICAL   OSCILLATIONS    OF 

I 

elongated,  by  &x-\  —     T^-™  —     ^^5  whence  dividing  by  A#, 

and  passing  to  the  limit,  we  obtain 

db1_1     P+(L-aQM 
dx~  KE 

Integrating  between  the  limits  0  and  L,  and  representing 
by  Lx  the  length  of  the  elongated  rod, 


If  the  strain  be  converted  into  a  thrust,  P  must  be  made 
to  assume  the  negative  sign;  and  if  this  thrust  equal  one 
half  the  weight  of  the  bar,  there  will  be  no  elongation  at  all. 


352.   THE   VERTICAL    OSCILLATIONS   OF   AN   ELASTIC   ROD    OK 
COED   SUSTAINING  A   GIVEN   WEIGHT   SUSPENDED   FROM   ITS 

EXTREMITY. 

Let  A  represent  the  point  of  suspension  of  the  rod  (fig.  1. 
on  the  next  page),  L  its  length  AB  before  its  elongation,  and 
\l  the  elongation  produced  in  it  by  a  given  weight  "W  sus- 
pended from  its  extremity,  and  C  the  corresponding  position 
of  the  extremity  of  the  rod. 

Let  the  rod  be  conceived  to  be  elongated  through  an 
additional  distance  CD=e  by  the  application  of  any  other 
given  strain,  and  then  allowed  to  oscillate  freely,  carrying 
with  it  the  weight  "W;  and  let  P  be  any  position  of  its 
extremity  during  any  one  of  the  oscillations  which  it  will 
thus  be  made  to  perform.  If,  then,  CP  be  represented  by  a?, 
the  corresponding  elongation  BP  of  the  rod  will  be  repre- 
sented by  iZ-ho?,  and  the  strain  which  would  retain  it  perma- 

KE 
nently  at  this  elongation  (equation  485)  by  -(i^+a?);  the 


unbalanced  pressure  or  moving  force  (Art.  92.)  upon  the 
weight  W,  at  the  period  of  this  elongation,  will  therefore  be 

Trnn  TT-p 

represented  by  -^-(%l+x)—  W,  or  by  -y-»;  since  W,  being 
the  strain  which  would  retain  the  rod  at  the  elongation  -J-Z,  is 

~ 


represented  by  -y~JZ  (equation  485). 

*  Whewell's  Analytical  Statics,  p.  113. 


A   LOADED    BAR. 


4:65 


The  unbalanced  pressure,  or  moving  force,  upon  the  mass 
"W  varies,  therefore,  as  the  distance  x  of  the  point  P  from  the 
given  point  0 ;  whence  it  follows  by  the  general  principle 
established  in  Art  97.,  that  the  oscillations  of  the  point  P 
extend  to  equal  distances  on  either  side  of  the  point  C,  as  a 
centre,  and  are  performed  isochronously,  the  time  T  of  each 
oscillation  being  represented  by  the  formula 


T-/WLU 

\JBSt 


.  (493). 


The  distance  from  A  of  the  centre  C,  about  which  th$> 
oscillations  of  the  point  P'  take  place,  is  represented  b$- 
L+-JZ;  so  that,  representing  this  distance  by  L0  and  substi- 
tuting for  %l  its  value,  we  have 


f 
i 


353'.  Let  us  now  suppose  that  when  in 
making  its  first  oscillation  about  C 
(fig.  2.)  the  weight  W  has  attained  its 
highest  position  d^  and  is  therefore,  for 
an  instant,  at  rest  in  that  position,  a 
second  weight  w  is  added  to  it ;  a  second 
series  of  oscillations  will  then  be  com- 
menced about  a  new  centre  C15  whose 
distance  L2  from  A  is  evidently  repre- 
sented by  the  formula 


So  that  the  distance  CO,  of  the  two  centres  is  ^—  ;  and  the 

ixiij 

greatest  distance  CJ)^  beneath  the  centre  0,,  attained  in  the 
second  oscillation,  equal  to  the  distance,  C^  at  which  the 
oscillation  commenced  above  that  point.  Now  C1DJ  = 


the  second  oscillation  is  therefore  %c  -f- 

30 


1=o+        ;  the  amplitude  df),  of 


4:66  THE  OSCILLATIONS   OF    A   LOADED   BAR. 

Let  the  weight  w  be  conceived  to  be  removed  when  the 
lowest  point  ~Dt  of  the  second  oscillation  is  attained,  a  third 
series  of  oscillations  will  then  be  commenced,  the  position  of 
whose  centre  being  determined  by  equation  (494),  is  identical 
with  that  of  the  centre  C,  about  which  the  first  oscillation 
was  performed.  In  its  third  oscillation  the  extremity  of  the 
rod  will  therefore  ascend  to  a  point  <#2  as  far  above  the  point 
C  as  D:  is  below  it  ;  so  that  the  amplitude  of  this  third  oscil- 

lation is  represented  by  2TCD,,  or  by  2CJD~TrCC!,  or  by 
^J).    When  the  highest  point  dz  of  this  third  oscil- 


lation  is  attained,  let  the  weight  w  be  again  added  ;  a  fourth 
oscillation  will  then  be  commenced,  the  position  of  whose 
centre  will  be  determined  by  equation  (495,)  and  will  there- 
fore be  identical  with  the  centre  C,,  about  which  the  second 
oscillation  was  performed  ;  so  that  the  greatest  distance  C,D? 
beneath  that  point  attained  in  this  fourth  oscillation  will  be 
equal  to  CA  or  to  CC^  -f  CDj  ;  and  its  amplitude  will  be 

represented  by  2  I  <?+__).    And  if  the  weight  w  be  thus 
\       K.Jii  / 

conceived  to  be  added  continually,  when  the  highest  point 
of  each  oscillation  is  attained,  and  taken  off  at  the  lowest 
point,  it  is  evident  that  the  amplitudes  of  these  oscillations 
will  thus  continually  increase  in  an  arithmetical  series  ;  so 
that  the  amplitude  A^  of  the  nili  oscillation  will  be  repre- 
sented by  the  formula 


The  ascending  oscillations  of  the  series  being  made  about 
the  centre  C,  and  the  descending  oscillations  about  C1? 
if  n  be  an  even  number,  the  centre  of  the  nih  oscillation  is 
Cj  ;  the  elongation  cn  of  the  rod  corresponding  to  the  lowest 
point  of  this  oscillation  is  therefore  equal  to  BQ+^A^  or 
substituting  for  BCj  its  value  given  by  equation  (495),  and 
for  An  its  value  from  equation  (496), 


Thus  it  is   apparent  that    by  the  long   continued   and 


DEFLEXION.  467 

periodical  addition  and  subtraction  of  a  weight  w,  so  small 
as  to  produce  but  a  slight  elongation  or  contraction  of  the 
rod  when  first  added  or  removed  from  it,  an  elongation  cn 
may  eventually  be  produced,  BO  great  as  to  pass  limits  of  its 
elasticity,  or  even  to  break  it.  Numerous  observations  have 
verified  this  fact:  the  chains  of  suspension  bridges  have 
been  broken  by  the  measured  tread  of  soldiers  ;*  and  M. 
Savart  has  shown,  that  by  fixing  an  elastic  rod  at  its  centre, 
and  drawing  the  wetted  finger  along  it  at  measured  inter- 
vals, it  may,  by  the  strain  resulting  from  the  slight  friction 
received  thus  periodically  upon  its  surface,  be  made  with 
great  ease  to  receive  an  oscillatory  movement  of  sufficient 
amplitude  to  be  measured,  f  M.  Poncelet  has  compared  the 
measurement  of  M.  Savart  with  theoretical  deductions 
analogous  to  those  of  the  preceding  article,  and  has  shown 
their  accordance  with  it. 


DEFLEXION. 
354.  The  neutral  surface  of  a  deflected  beam. 

One  surface  of  a  beam  becoming,  when  deflected,  convex, 
and  the  other  concave,  it  is  evident  that  the  material  form- 
ing that  side  of  the  beam  which  is  bounded  by  the  one 
surface  is,  in  the  act  of  flexure,  extended,  and  that  of  the 
other  compressed.  The  surface  which  separates  these  two 
portions  of  the  material  being  that  where  its  extension  ter- 
minates and  its  compression  begins,  and  which  sustains, 
therefore,  neither  extension  nor  compression,  is  called  the 
NEUTRAL  SURFACE. 

355.  THE  POSITION  OF  THE  NEUTRAL  SURFACE  OF  A  BEAM. 
Let  ABCD  be  taken  to  represent  any  thin  lamina  of  the 

*  Such  was  the  fate  of  the  suspension  bridge  at  Broughton  near  Manchester, 
the  circumstances  of  which  have  been  ably  detailed  by  Mr.  E.  Hodgkinson  in 
the  fourth  volume  of  the  Manchester  Philosophical  Transactions.  M.  Navier 
has  shown,  in  his  treatise  on  the  theory  of  suspension  bridges  (Sur  les  Fonts 
Suspendus,  Paris,  1823),  that  the  duration  of  the  oscillations  of  the  chains  of 
a  suspension  bridge  may  in  certain  cases  extend  to  nearly  six  seconds  ;  there 
might  easily,  in  such  cases,  arise  that  isochronism  at  each  interval,  or  after 
any  number  of  intervals,  between  the  marching  step  of  the  troops  and  the 
oscillations  of  the  bridge,  whence  would  result  a  continually  increasing  elon- 
gation of  the  suspending  chains. 

f  Mecanique  Industrielle,  p.  437,  Art.  331. — ED. 


468  THE  NEUTRAL  SURFACE 

beam  contained  by  planes 
parallel  to  the  plane  of  its 
deflexion,  and  P1?  P,,  P8  the 
resultants  of  all  the  pres- 
sures applied,  to  it ;  acb  that 
portion  of  the  neutral  sur- 
iace  of  the  beam  which  is 
contained  within  this  la- 
mina, and  may  be  called  its 
neutral  line;  PT  and  QY 
planes  exceedingly  near  to 
one  another,  and  perpen- 
dicular to  the  neutral  line  at  the  points  where  they  intersect 
it ;  and  O  the  intersection  of  PT  and  QY  when  produced. 

Now  let  it  be  observed  that  the  portion  ArTD  of  the 
beam  is  held  in  equilibrium  by  the  resultant  pressure  P1? 
and  by  the  elastic  forces  called  into  operation  upon  the  sur- 
face PT  ;  of  which  elastic  forces  those  acting  in  PR  (where 
the  material  of  the  beam  is  extended)  tend  to  bring  the 
points  to  which  they  are  severally  applied  nearer  to  the 
plane  SQ,  and  those  acting  in  RT  (where  the  material  is 
compressed),  to  carry  their  several  points  of  application 
farther  from  the  plane  SY. 

Let  aR=x,  SR=A#,  and  imagine  the  lamina  PQYT  to  be 
made  up  of  fibres  parallel  to  SR ;  then  will  Aa?  represent 
the  length  of  each  of  these  fibres  before  the  deflexion  of  the 
beam,  since  the  length  of  the  neutral  fibre  SR  has  remained 
unaltered  by  the  deflexion.  Let  dx  represent  the  quantity 
by  which  the  fibre  pq  has  been  elongated  by  the  deflexion 
of  the  beam,  then  is  the  actual  length  of  that  fibre  repre- 
sented by  AaH-cSa?.  Whence  it  follows  (equation  485),  that 
the  pressure  which  must  have  operated  to  produce  this 

dx 

elongation  is  represented  by  E— A&,  A&  being  taken  to  repre- 
sent the  section  of  the  fibre,  or  an  exceedingly  small  element 
of  the  section  PT  of  the  lamina.  Now  PT  and  QY  being 
normals  to  SR,  the  point  O  in  which  they  meet,  when 
produced,  is  the  centre  of  curvature  to  the  neutral  line  in 
R.  Let  the  radius  of  curvature  OR  be  represented  by  R, 

Op 
and  the  distance  ~Rp  by  p.      By  similar  triangles,  Q^= 

pq          R-fp     Aa?+da?  p  <fo     Al       « 

^jL.j  or  __/== 9  or  !+:£-=  1  + —  ;  therefore. 


p 


OF  A   BEAM.  469 

—  .     Substituting  this  value  of  —  in  the  expression  for  the 

pressure  which  must  have  operated  to  produce  the  elonga- 
tion of  the  fibre  pq,  and  representing  that  pressure  by  AP, 
we  have 

AP=*|A&  ____  (498). 

If,  therefore,  KP  be  represented  by  ^  and  ET  by  &,,  then 
the  sum  of  the  elastic  forces  developed  by  the  extension  of 

E  kl 
the  fibres  in  RPQS  is  represented  by  ^-20pA&;  and,  similarly, 

the  sum  of  those  developed  by  the  compression  of  the  fibres 

E  *2 
in  RTVS  is  represented  by  =-20pA&.     Now  let  it  be  observed 

that  (since  the  pressures  applied  to  APTD,  and  in  equili- 
brium, are  the  forces  of  extension  and  compression  acting 
in  KP  and  RT  respectively,  and  the  pressure  PJ,  if  the 
pressure  Px  be  resolved  in  a  direction  perpendicular  to  the 
plane  PT,  or  parallel  to  the  tangent  to  the  neutral  line  in  R, 
this  resolved  pressure  will  be  equal  (Art.  16.)  to  the  differ- 
ence of  the  sums  of  the  forces  of  extension  and  compression 
applied  (in  directions  perpendicular  to  that  plane,  but  oppo- 
site to  one  another)  to  the  portions  RP  and  RT  of  it  respec- 
tively. Representing,  therefore,  by  6  the  inclination  R0P, 
of  the  direction  of  Pj  to  the  normal  to  the  neutral  line  in  R, 
we  have 

~p*  kl 

Pa  sin.  4=-20pA&_ 

JK.  K 

But  if  Jc  be  taken  to  represent  the  whole  section  PT,  and  h 
the  distance  of  the  point  R  from  its  centre  of  gravity,  then 
(Art.  18.) 

.-.  P,  sin.  4= 


RP 
...^-^sin.4  .....  (499); 

which  expression  represents  the  distance  of  the  neutral  line 
from  the  centre  of  gravity  of  any  section  PT  of  the  lamina, 
that  distance  being  measured  towards  the  extended  or  the 
compressed  side  of  the  lamina  according  as  d  is  positive  or 


470 


RADIUS   OF   CURVATURE. 


negative;  so  that  the  neutral  line  passes  from  one  side  to 
the  other  of  the  line  joining  the  centres  of  gravity  of  the 
cross  sections  of  the  lamina,  at  the  point  where  d— 0,  or  at 
the  point  where  the  normal  to  the  neutral  line  is  parallel  to 
the  direction  of  P. 


356.  Case  of  a  rectangular  beam. 

If  the  form  of  the  beam  be  such  that  it  may  be  divided 
into  laminae  parallel  to  ABCD  of  similar  forms  and  equal 
dimensions,  and  if  the  pressure  P,  applied  to  each  lamina 
may  be  conceived  to  be  the  same  ;  or  if  its  section  be  a  rec- 
tangle, and  the  pressures  applied  to  it  be  applied  (as  they 
usually  are)  uniformly  across  its  width,  then  will  the  distance 
h  of  the  neutral  line  of  each  lamina  from  the  centre  of  gra- 
vity of  any  cross  section  of  that  lamina,  such  as  PT,  be  the 
same,  in  respect  to  corresponding  points  of  all  the  laminae, 
whatever  may  be  the  deflection  of  the  beam ;  so  that  in  thig 
case  the  neutral  surface  is  always  a  cylindrical  surface. 


357.  Case  in  which  the  deflecting  pressure  P1  is  nearly  per- 
pendicular to  the  length  of  the  learn. 

In  this  case  0,  and  therefore  sin.  0,  is  exceeding  small,  so 
long  as  the  deflexion  is  small  at  every  point  R  of  the  neutral 
line ;  so  that  h  is  exceedingly  small,  and  the  neutral  line 
of  the  lamina  passes  very  nearly,  or  accurately,  through  the 
centre  of  gravity  of  its  section  PT. 


358.  THE  RADIUS   OF   CURVATURE   OF  THE  NEUTRAL  SURFACE 

OF   A  BEAM. 

Since  the  pressures  applied  to  the  portion  APTD  of  the 


lamina  ABCD  are  in  equilibrium,  the  principle  of  the  equality 
of  moments  must  obtain  in  respect  to  them  ;   taking,  there- 


RADIUS   OF   CURVATURE.  471 

fore,  the  point  R,  where  the  neutral  axis  of  the  lamina  inter- 
sects PT,  as  the  point  from  which  the  moments  are  measured, 
and  observing  that  the  elastic  pressures  developed  by  the 
extension  of  the  material  in  RP  and  its  compression  in  RT 
both  tend  to  turn  the  mass  APTD  in  the  same  direction 
about  the  point  R,  and  that  each  such  pressure  upon  an 

element  &k  of  the  section  PT  is  represented  (equation  498)  by 
• 


_  pA&5  and  therefore  the  moment  of  that  pressure  about  the 
R 

E 

point  R  by  —  paA&,  it  follows  that  the  sum  of  the  moments 

R 
about  the  point  R  of  all  these  elastic  pressures  upon  PT  is 

represented  by  =  -2p2A&5  or  by  --,  if  I  be  taken  to  represent 

R  R 

the  moment  of  inertia  of  PT  ^bout  R.  Observing,  moreover, 
that  if  p  represent  the  length  of  the  perpendicular  let  fall 
from  R  upon  the  direction  of  any  pressure  P  applied  to  the 
portion  APTD  of  the  beam,  fp  will  represent  its  moment, 
and  SPp  will  represent  the  sum  of  the  moments  of  all  the 
similar  pressures  applied  to  that  portion  of  the  beam  ; 
we  have  by  the  principle  of  the  equality  of  moments, 


359.  The  neutral  surface  of  the  beam  is  a  cylindrical  sur- 
face, whatever  may  be  its  deflection  or  the  direction  of  its 
deflecting  pressure,  provided  that  its  section  is  a  rectangle 
(Art.  356.) ;  or  whatever  may  be  its  section,  provided  that  its 
deflection  be  small,  the  direction  of  the  deflecting  pressure 
nearly  perpendicular  to  its  length,  and  its  form  before  de- 
flexion symmetrical  in  respect  to  a  plane  perpendicular  to  the 
plane  of  deflexion.  In  every  such  case,  therefore,  the  neutral 
lines  of  all  the  laminae  similar  to  ABCD,  into  which  the 
beam  may  be  divided,  will  have  equal  radii  of  curvature  at 
points  similar  to  R  lying  in  the  same  right  line  perpendicular 
to  the  plane  of  deflection ;  taking,  therefore,  equations  simi- 
lar to  the  above  in  respect  to  all  the  laminae,  multiplying 
both  sides  of  each  by  I,  adding  them  together,  and  observ- 
ing that  R  and  E  are  the  same  in  all,  we  have  _  _  Pm 

R         E 
In  this  case,  therefore,  I  may  be  taken  in  equation  (COO)  to 


472  MOMENT   OF  INERTIA. 

represent  the  moment  of  inertia  of  the  whole  section  of  the 
beam,  and  P  the  pressure  applied  across  its  whole  width. 


360.  The  radius  of  curvature  of  a  beam  whose  deflexion  is 
small,  and  the  direction  of  me  deflecting  pressures  nearly 
perpendicular  to  the  length  of  the  beam. 

In  this  case  the  neutral  line  is  very  nearly  a  straight  line, 
perpendicular  to  the  directions  of  the  deflecting  pressures  ; 
so  that,  representing  its  length  by  a?,  we  have,  in  this  case, 
jp=x  ;  and  equation  (500)  becomes 


which  relation  obtains,  whatever  may  be  the  form  of  the 
transverse  section  of  the  beam,  I  representing  its  moment  of 
inertia  in  respect  to  an  axis  passing  through  its  centre  of 
gravity  and  perpendicular  to  the  plane  of  deflexion. 


361.  The  moment  of  inertia  I  of  the  transverse  section  of  a 
beam  about  the  centre  of  gravity  of  the  section. 

In  treating  of  the  moments  of  inertia  of  bodies  of  different 
geometrical  forms  in  a  preceding  part  of  this  work  (Art.  82, 
&c.),  we  have  considered  them  as  solids ;  whereas  the  mo- 
ment of  inertia  I  of  the  section  of  a  beam  which  enters  into 
equation  (500)  and  determines  the  curvature  of  the  beam 
when  deflected,  is  that  of  the  geometrical  area  of  the  section. 
Knowing,  however,  the  moment  of  inertia  of  a  solid  about 
any  axis,  whose  section  perpendicular  to  that  axis  is  of  a 
given  geometrical  form,  we  can  evidently  determine  the 
moment  of  the  area  of  that  section  about  the  same  axis,  by 
supposing  the  solid  in  the  first  place  to  become  an  exceed- 
ingly thin  lamina  (i.  e.  by  making  that  dimension  of  the 
solid  which  is  parallel  to  the  axis  exceedingly  small  in  the 
expression  for  the  moment  of  inertia),  and  then  dividing 
the  resulting  expression  by  the  exceedingly  small  thickness 
of  this  lamina.  We  shall  thus  obtain  the  following  values 
of  I:- 


MOMENT   OF   INERTIA.  473 

362.    For  a  beam  with  a  rectangular  section,  ) 
whose  breadth  is  represented  by  b  and  its  depth  V&a 
by  G  (equation  61),  ) 


363.  For  a  beam  with   a  triangular  )          ic 
section,  whose  base  is  b  and  its  height  c  >  I  =  ~^r 
(equation  63),  ) 

364.  For  a  beam  or  column  with  a  circular ) 
section,  whose  radius  is  c  (equation  66),  j 


365.  To  determine  the  moment  of  inertia  I  in  respect  to  a 
A  B       beam  whose  transverse  section  is  of  the 

1  —  n  f  -  '  form  represented  in  the  accompanying 
figure,  about  an  axis  ab  passing  through 
its  centre  of  gravity  ;  let  the  breadth  of 
the  rectangle  AB  be  represented  by  bl  and 
its  depth  by  d^  and  let  &2  and  dz  be  simi- 
larly taken  in  respect  to  the  rectangle  EF, 
and  ba  and  d3  in  respect  to  CD  ;  also  let  It 
represent  the  moment  of  inertia  of  the  section  about  the  axis 
cd  passing  through  the  centre  of  CD,  A1?  A,,  A,,  the  areas 
of  the  rectangles  respectively,  and  A  the  area  of  the  whole 
section. 

Now  the  moments  of  inertia  of  the  several  rectangles, 
about  axes  passing  through  their  centres  of  gravity,  are 
represented  by  yV^A'i  A^A3?  rV^AN  an(^  tne  distances  of 
these  axes  from  the  axis  cd  are  respectively 
0.  Therefore  (equation  58), 


but  A,=M»  A,=M»  AS= 


Also  if  h  represent  the  distance  between  the  axes  ab  and  cd, 
then  (Art.  18)  AA=i(da  +  rf,)Aa—  i^  +  ^^o  and  (equation 

58)  Irnl.-A'A. 


If  dl  and  d^  be  exceedingly  small  as  compared  with  <28, 


4T4 


DEFLEXION    OF  A    BEAM. 


neglecting  their  values  in  the  two  last  terms  of  the  equation 
and  reducing,  we  obtain 


(503). 

If  the  areas  AB  and  EF  be  equal  in  every  respect, 
1=1  {d* +  *&  +  %)*}  At  +  JsAj;  .....  (504). 


366.  THE  WORK  EXPENDED  UPON  THE  DEFLEXION  OF  A  BEAM 
TO  WHICH  GIVEN  PRESSURES  ARE  APPLIED. 

If  AP  represent  the  pressure  which  must  have  operated 

to  produce  the  elongation  or 
compression  which  the  ele- 
mentary fibre  pq  receives, 
by  reason  of  the  deflexion 
of  the  beam,  AOJ  the  length 
of  that  fibre  before  the  de- 
flexion of  the  beam,  and  &k 
its  section;  then  the  work 
which  must  have  been  done 
upon  it,  thus  to  elongate 
or  compress  it,  is  repre- 
sented, equation  (487)  by 

498)  Ap=&4jfc 


E    Aft' 

pended  upon  the  extension  or  compression  of  pq  is  there- 
fore represented  by 


And  the  same  being  true  of  the  work  expended  on  the 
compression  or  extension  of  every  other  fibre  composing  the 
elementary  solid  VTPQ,  it  follows  that  the  whole  work 
expended  upon  the  deflexion  of  that  element  of  the  beam 

is  represented  by  J-^j-  2p3A&,  or  by  i™A#  5  for  2p2A&  repre- 
sents the  moment  of  inertia  I  of  the  section  PT,  about  an 
axis  perpendicular  to  the  plane  of  ABCD,  and  passing  through 
the  point  K.  If,  therefore,  0t  be  taken  to  represent  the 
length  of  that  portion  of  the  beam  which  lies  between  D 


DEFLEXION   OF  A  BEAM.  475 

and  M  before  its  deflexion,  and  therefore  the  length  of  the 
portion  ac  of  its  neutral  line  after  deflexion,  then  the  whole 
work  expended  upon  the  deflexion  of  the  part  AM  of  the 

01 1  1 

beam  is  represented  lyy  %  E2    ^-2Aa?.   But  (equation  500)  ^= 

o-tv  J*> 

yara"  j    whence,    by    substitution,    the    above    expression 

P  22«i# a 

becomes  j~4r  o  y  Aa?-  Passing  to  the  limit,  and  represent- 
ing the  work  expended  upon  the  deflexion  of  the  part  AM 
of  the  beam  by  u^ 

P,a 


o 


367.  TA0  work  expended  upon  the  deflexion  of  a  learn  of 
uniform  dimensions,  when  the  deflecting  pressures  are 
nearly  perpendicular  to  the  surface  of  the  beam. 

In  this  case  I  is  constant,  and  ^=8?;  whence  we  obtain 
by  integrating  (equation  505)  be- 
tween the  limits  0  and  a1? 

u>=^if  ••••  (506)> 

where  uv  represents  the  work  ex- 
pended upon  the  deflexion  of  the 
portion  AM  of  the  beam.  Simi- 
larly, if  bc=a^  the  work  expended 

upon  the  deflexion  of  the  portion  BM  of  the  beam  is  repre- 

sented by 


so  that  the  whole  work  Us  expended  upon  the  deflexion  of 
the  beam  is  represented  by 


P2//  8 
\a\ 


6EI 


But  by  the  principle  of  the   equality  of  moments,  if   a 
represent  the  whole  length  of  the  beam, 


476 


DEFLEXION   OF  A   BEAM. 


Eliminating  P1  and  Pa  between  these  equations  and  the  pre- 
ceding, we  obtain  by  reduction 


(507). 


If  the  pressure  P3  be  applied  in  the  centre  of  the  beam, 


368.  THE  LINEAR  DEFLEXION  OF  A  BEAM  WHEN  THE  DIRECTION 

OF     THE     DEFLECTING     PRESSURE     IS     PERPENDICULAR     TO    ITS 
SURFACE. 


Let 


the  section  MK  remain  fixed,  the  deflexion  taking 
place  on  either  side  of  that  section  ; 
then  u^  representing  the  work  ex- 
pended upon  the  deflexion  of  the 
portion  AM  of  the  beam,  and  D1 
the  deflexion  of  the  point  to  which 
P!  is  applied,  measured  in  a  direc- 
tion perpendicular  to  the  surface,  we 

(e(luatio11  40)>  ^i 


du,       du,      dP* 
' 


, 
therefore    P.  =  -~r-  — 


But  by  equation  (506),         =  l         ;  therefore  P,  - 


-jf^  5  therefore  -^-  =  ^  Wy  ;  whence  we  obtain  by  integration 


3EI 


(509). 


If  the  whole  work  of  deflecting  the  beam  be  done  by  the 
pressure  P3,  the  points  of  application  of  P  and  P2  having  no 
motions  in  the  directions  of  these  pressures  (Art.  52.),  then 
proceeding  in  respect  to  equation  (507)  precisely  as  before  in 
respect  to  equation  (506),  and  representing  the  deflexion 


*  Church's  Diff.  Cal.    Art.  1Y. 


DEFLEXION  OF   A  BEAM.  477 

perpendicular  to  the  surface  of  the  beam  at  the  point  of 
application  of  P3  by  D3,  we  shall  obtain 


If  the  pressure  P8  be  applied  at  the  centre  of  the  beam 


Eliminating  P,  between  equations  (506)  and  (509),  and  P, 
between  equations  (507)  and  (510),  we  obtain 


by  which  equations  the  work  expended  upon  the  deflexion 
of  a  beam  is  determined  in  terms  of  the  deflexion  itself,  as 
by  equations  (506)  and  (507)  it  was  determined  in  terms  of 
the  deflecting  pressures. 


369.  CONDITIONS  OF  THE  DEFLEXION  OF  A  BEAM  TO  WHICH  ARE 
APPLIED  THREE  PRESSURES,  WHOSE  DIRECTIONS  ARE  NEARLY 

PERPENDICULAR   TO   ITS    SURFACE. 

Let  AB  represent  any  lamina  of  the  beam  parallel  to  its 

H 


plane  of  deflexion,  and  acb  the  neutral  line  of  that  lamina 
intersected  by  the  direction  of  P8  in  the  point  c. 

Draw  xx1  parallel  to  the  length  of  the  beam  before  its 
deflexion,  and  take  this  line  as  the  axis  of  the  abscissae,  and 
the  point  o  as  the  origin ;  then,  representing  by  x  and  y  the 

*  This  result  is  identical  with  that  obtained  by  a  different  method  of  inves- 
tigation by  M.  Navier  (Resume  de  Lemons  de  Construction,  Art.  359.). 


4:78  EQUATION   TO   THE   NEUTRAL   LINE. 

co-ordinates  of  any  point  in  ac,  and  by  ~R  the  radius  of  curva- 
ture of  that  point,  we  have  * 


Now  the  deflexion  of  the  beam  being  supposed  exceed- 
ingly small,  the  inclination   to   ex  of  the  tangent   to  the 

neutral  line  is,  at  all  points,  exceedingly  small,  so  that  I  -^J 

may  be  neglected  as  compared  with  unity  ;  therefore  ^—  -^. 

Substituting  this  value  in  equation  (501),  and  observing  that 
in  this  casejp  is  represented  by  (a,—  a?)  instead  of  #, 

(513). 

the  direction  of  the  pressure  Pa  being  supposed  nearly  per 
pendicular  to  the  surface  of  the  beam,  and  I  constant.  Let 
the  above  equation  be  integrated  between  the  limits  0  and 
a?,  /3  being  taken  to  represent  the  inclination  of  the  tangent 

at  c  to  «B,  so  that  the  value  of  f  at  c  may  be  represented  by 

GuX 

tan.  13, 

-tan./SXo,-^   ...(514). 


Integrating  a  second  time  between  the  limits  0  and  a?,  and 
observing  that  when  a?=0,  y=0, 

y=^l\fatf-ia?}+a>teai.P  ....  (515). 

Proceeding  similarly  in  respect  to  the  portion  ~bc  of  the  neu- 
tral line,  but  observing  that  in  respect  to  this  curve  the  value 

of  —  -     at  the  point  c  is  represented  by  tan.  /3,  we  have 


dx*~~      El 
./3==^M^-^}   .  .  (516). 

£jL 

Church's  Diff.  Cal.     Art.  105. 


EQUATION   TO   THE   NEUTRAL   LINE.  479 


If  Dl  and  D2  be  taken  to  represent  the  deflexions  at  the 
points  a  and  £,  and  ca  and  cb  be  assumed  respectively  equal 
to  cd  and  ce, 

by  equation  (515),  D1=    l-A  +gt  tan.  /3, 

P  #  3 
by  equation  (517),  D5=—  ^~—  «2  tan.  £. 


If  the  pressures  P1  and  P2  be  supplied  by  the  resistances 
of  fixed  surfaces,  then  T>l=T>,l.  Subtracting  the  above  equa- 
tion we  obtain,  on  this  supposition, 


tan. 


Now   FA'-PA'=     ''-~'=P8a.^(gl-a.)  ;  ob- 
serving that  P1a=P,<*,,  ?,»=?,»„  and  «,+«,:=  a, 


If  /315  /32  represent  the  inclinations  of  the  neutral  line  to 
xxl  at  the  points  a  and  J,  then  by  equations  (514)  and  (516) 

tan./3,_tan./3=,  tan.  ^ 


Substituting  for  tan.  /3  its  value  from  equation  (518),  elimi- 
nating and  reducing, 

_Pia.g,(gI  +  2g.)  .  _P.g, 

1D-  ^1-  a'  ^2" 


To  determine  the  point  m  where  the  tangent  to  the  neutral 
line  is  parallel  to  cxx^  or  to  the  undeflected  position  of  the 

fii  i 

beam,  we  must  assume  -/-=0  in    equation   (516)*  ;    if   we 

fftx 

then  substitute  for  tan.   (3  its  value   from  equation    (518), 
substitute  for  P2  its  value  in  terms  of  P3,  and  solve  the 

*  Church's  Diff.  Cal.     Art.  78. 


480 


LENGTH   OF   THE   NEUTRAL   LINE. 


resulting  equation  in  respect  to  a?,  we  shall  obtain  for  the 
distance  of  the  point  m  from  c  the  expression 


370.  THE  LENGTH  OF  THE  NEUTRAL  LINE,  THE  BEAM  BEING 
LOADED  IN  THE  CENTRE. 

Let  the  directions  of  the  resistances  upon  the  extremities 


of  the  beam  be  supposed  nearly  perpendicular  to  its  surface  ; 
then  if  x  and  y  be  the  co-ordinates  of  the  neutral  line  from 
the  point  #,  we  have  (equation  501),  representing  the  hori- 
zontal distance  AB  by  20,  and  observing  that  in  this  case 

-=—-5-5,  and  that  the  resistance  at  A  or  B  =  JP, 


Integrating  between  the  limits  x  and  0,  and  observing  that 

d/u 
at  the  latter  limit  -     =  Or 


Now  if  *  represent  the  length  of  the  curve  ac, 


Church's  Int.  Cal.     Art.  197. 


THE   DEFLEXION    OF   A   BEAM. 


481 


the  deflexion  being  small,  -T->  is  exceedingly  small  at  every 
point  of  the  neutral  line. 


:.a  =  f  | 


'.*  =  «+  60ET-...  (521). 


Eliminating  P  between  this  equation  and  equation  (511),  and 
representing  the  deflexion  by  D, 


Da* 

-. 


371.  A  BEAM,  ONE  PORTION  OF  WHICH  IS  FIRMLY  INSERTED  IN 
MASONRY,  AND  WHICH  SUSTAINS  A  LOAD  UNIFORMLY  DISTRI- 
BUTED OVER  ITS  REMAINING  PORTION. 

Let  the  co-ordinates  of  the  neutral  line  be  measured  from 


*  The  following  experiments  were  made  by  Mr.  Hatcher,  superintendant  of 
the  work-shop  at  King's  College,  to-verify  this  result,  which  is  identical  with 
that  obtained  by  M.  Navier  (Resume  des  Lemons,  Art.  86.).  Wrought  iron 
rollers  -7  inch  in  diameter  were  placed  loosely  on  wrought  iron  bars,  the  sur- 
faces of  contact  being  smoothed  with  the  file  and  well  oiled.  The  bar  to  be 
tested  had  a  square  section,  whose  side  was  '7  inch,  and  was  supported  on  the 
two  rollers,  which  were  adjusted  to  10  feet  apart  (centre  to  centre)  when  the 
deflecting  weight  had  been  put  on  the  bar.  On  removing  the  weights-care- 
fully,  the  distance  to  which  the  rollers  receded  as  the  bar  recovered  its  hori- 
zontal position  was  noted. 


Deflecting  Weight 

Deflection  in  Inches. 

Distance  through  .which 
each  Roller  receded 
in  inches. 

Distance  through  which 
each  Roller  would  have 
receded  by  Formula.  . 

56 
84 

3-7 
5-45 

•1 

•2 

•13 
•29 

31 


THE   DEFLEXION   OF   A   BEAM 


the  point  B  where  the  beam 
is  inserted  in  the  masonry, 
and  let  the  length  of  the 
portion  AD  which  sustains 
the  load  be  represented  by 
#,  and  the  load  upon  each 
unit  of  its  length  by  p  ; 
then,  representing  by  x  and 
y  the  co-ordinates  of  any 
point  P  of  the  neutral  line, 
and  observing  that  the  pres- 
sures applied  to  AP,  and  in 

equilibrium,  are  the  load  ^(a—x)  and  the  elastic  forces 
developed  upon  the  transverse  section  at  P,  we  have  by  the 
principle  of  the  equality  of  moments,  taking  P  as  the  point 
from  which  the  moments  are  measured,  and  observing  that 
since  the  load  p(a—x)  is  uniformly  distributed  over  AP  it 
produces  the  same  effect  as  though  it  were  collected  over  the 
centre  of  that  line,  or  at  distance  \(a—  x)  from  P  ;  observing, 
moreover,  that  the  sum  of  the  moments  of  the  elastic  forces 
upon  the  section  at  P,  about  that  point,  is  represented  (Art. 

358.)  by  5  or  by  El  J  (Art.  369.)  ; 


Integrating  twice  between  the  limits  0  and  a?,  and  observing 

CM  i  V/ 

that  when  x=Q,-^-=Q  and  y=0,  since  the  portion  BC  of  the 
beam  is  rigid,  we  obtain 


which  is  the  equation  to  the  neutral  line. 

Let,  now,  a  be  substituted  for  x  in  the  above  equation  ; 
and  let  it  be  observed  that  the  corresponding  value  of  y 
represents  the  deflexion  D  at  the  extremity  A  of  the  beam  ; 
we  shall  thus  obtain  by  reduction 


LOADED   UNIFORMLY. 


483 


Representing  by  P  the  inclination  to  the  horizon  of  the  tan- 
gent to  the  neutral  line  at  A,  substituting  a  for  x  in  equation 

(523),  and  observing  that  when  x=a,  -/^=  tan.  /8,  we  obtain 
tan.  /3=|^  .  .  .  .  (526). 


372.    A  BEAM   SUPPORTED   AT   ITS   EXTREMITIES   AND   SUSTAINING 
A   LOAD   UNIFORMLY   DISTRIBUTED   OVER   ITS    LENGTH. 

Let  the  length  of  the  beam  be  represented  by  20,  the  load 
upon  each  unit  of  length  by  p  ;  take 
x  and  y  as  the  co-ordinate  of  any 
point  P  of  the  neutral  line,  from  the 
origin  A;  and  let  it  be  observed 
that  the  forces  applied  to  AP,  and  in 
equilibrium,  are  the  load  px  upon  that 
portion  of  the  beam,  which  may  be 
supposed  collected  over  its  middle 
point,  the  resistance  upon  the  point  A,  which  is  represented 
by  pa,  and  the  elastic  forces  developed  upon  the  section 
atP;  then  by  Art.  360., 


Integrating  this  equation  between  the  limits  x  and  #,  and 

observing  that  at  the  latter  limit  -^  =  0,  since  y  evidently 

dx 
attains  its  maximum  value  at  the  middle  C  of  the  beam, 

=  ifx(^-«8)-4K«a-«9)  ____  (528). 


Integrating  a  second  time  between  the  limits  0  and  a?,  and 
observing  that  when  oj*=0,  y=0, 


3-a'x)  ____  (529), 
which  is  the  equation  to  the  neutral  line.     Substituting  a  for 


THE   DEFLEXION   OF   A  BEAM 


x  in  this  equation,  and  observing  that  the  corresponding 
value  of  y  represents  the  deflexion  D  in  the  centre  of  the 
beam,  we  have  by  reduction 


Representing  by  /3  the  inclination  to  the  horizon  of  the  tan- 
gent to  the  neutral  line  at  A  or  B,  and  observing  that  when 

a?=0  in  equation  (528),^=  tan.  /3, 
ax 

=      L  .....  (531). 


Let  it  be  observed  that  the  length  of  the  beam,  which  in 
equation  (511)  is  represented  by  #,  is  here  represented  by 
2w,  and  that  equation  (530)  may  be  placed  under  the  form 

D=j-  .      AQ-FT     ?  whence  it  is  apparent  that  ihe  deflexion 


of  a  beam,  when  uniformly  loaded  throughout,  is  the  same 
as  though  -fths  of  that  load  (2^)  were  suspended  from  its 
middle  point. 


373.  A  BEAM  IS  SUPPORTED  BY  TWO  STRUTS  PLACED  SYM- 
METRICALLY, AND  IT  IS  LOADED  UNIFORMLY  THROUGHOUT 
ITS  WHOLE  LENGTH;  TO  DETERMINE  ITS  DEFLEXION. 

Let  CD=2&,  CA  =  a1,  load  upon  each  foot  of  the  length 
of  the  beam^fx ;  then  load  on 
each  point  of  support =v-a.  Take 
C  as  the  origin  of  the  co-ordinates ; 
then,  observing  that  the  forces 
impressed  upon  any  portion  CP 
of  the  beam,  terminating  between 
C  and  A,  are  the  elastic  forces 
upon  the  transverse  section  of  the 
beam  at  P,  and  the  weight  of  the 
load  upon  CP ;  and  observing  that  the  weight  M-CP  of  the 
load  upon  CP,  produces  the  same  effect  as  though  it  were 
collected  over  the  centre  of  that  portion  of  the  beam,  so  that 
its  moment  about  the  point  P  is  represented  by  p.  CP. 


LOADED   UNIFORMLY.  485 

or  by  -|^CPa  ;  we  obtain  for  the  equation  to  the  neutral  line 
in  respect  to  the  part  CA  of  the  beam  (Art.  360) 


(532). 


Since,  moreover,  the  forces  impressed  upon  any  portion  CQ 
of  the  beam,  terminating  between  A  and  E,  are  the  elastic 
forces  developed  upon  the  transverse  section  at  Q,  the 
resistance  pa  of  the  support  at  A,  and  the  load  upon  CQ, 
whose  moment  about  Q  is  represented  by  J-^CQ2,  we  have 
(equation  501),  representing  CQ  by  a?, 


EI      =f(«e°-Ka'-0,)  .....  (533). 

Representing  the  inclination  to  the  horizon  of  the  tangent  to 
the  neutral  line  at  A  by  /3,  dividing  equation  (532)  by  v*, 
integrating  it  between  the  limits  x  and  a15  and  observing 

ffll 

that  at  the  latter  limit  -^=tan.  /3,  we  have,  in  respect  to  the 
portion  CA  of  the  beam, 


Integrating  equation  (533)  between  the  limits  x  and  «,  and 
observing  that  at  the  latter  limit  ^=sO,  since  the  neutral 
line  at  E  is  parallel  to  the  horizon, 

—  -7-= |#s— \a(x— a^— ±a*+%a(a— a$ (535); 

|X     (JUvu 

which  equation  having  reference  to  the  portion  AE  of  the 

beam,  it  is  evident  that  when  #=#.,  -y-=tan.  P. 

IJ  dx 

•  —        8-1 

....  (536). 

Substituting,  therefore,  for  tan.  P  in  equation  (534),  and 
reducing,  that  equation  becomes 


486  THE   DEFLEXION    OF   A   BEAM 

Integrating  equation  (535)  between  the  limits  a,  and  a?,  and 
equation  (537)  between  the  limits  0  and  a?,  and  representing 
the  deflexion  at  C,  and  therefore  the  value  of  y  at  A,  by  D0 

El 

—  (y-DO 


(538); 


the  former  of  which  equations  determines  the  neutral  line 
of  the  portion  AE,  and  the  latter  that  of  the  portion  CA  of 
the  beam.  Substituting  at  for  x  in  the  latter,  and  observing 
that  y  then  becomes  D:  ;  then  substituting  this  value  of  D1 
in  the  former  equation,  and  reducing, 


'-&-ad\  ••••  (539); 

El 


ay-c?}x  ----  (540). 

f** 

The  latter  equation  being  that  to  the  neutral  line  of  the  por- 
tion AE  of  the  beam,  if  we  substitute  a  in  it  for  a?,  and 
represent  the  ordinate  of  the  neutral  line  at  E  by  y^  we 
shall  obtain  by  reduction 

<*-«,)°-3«s5  ....  (541). 

If  a^O,  or  if  the  loading  commence  at  the  point  A  of  the 
beam,  the  value  of  y^  will  be  found  to  be  that  already  deter- 
mined for  the  deflexion  in  this  case  (equation  530). 

Now,  representing  the  deflexion  at  E  by  D2,  we  have  evi- 
dently D2—  Dx—  yr 


(542). 


374.  THE  CONDITIONS  OF  THE  DEFLEXION  OF  A  BEAM  LOADED 
UNIFORMLY  THROUGHOUT  ITS   LENGTH,  AND   SUPPORTED   AT 

ITS  EXTREMITIES  A  AND  D,  AND  AT  TWO  POINTS  B  AND  C 
SITUATED  AT  EQUAL  DISTANCES  FROM  THEM,  AND  IN  THE 
SAME  HORIZONTAL  STRAIGHT  LINE. 


Let  AB=^,  AD  =20. 

Let  A  be  taken  as  the  origin  of  the  co-ordinates  ;  let  the 


LOADED   UNIFORMLY. 


487 


pressure  upon  that  point  be 
represented  by  P15  and  the 
pressure  upon  B  by  P2  ;  also 
the  load  upon  each  unit  of 
the  length  of  the  beam  by  p. 
If  F  be  any  point  in  the 
neutral  line  to  the  portion  AB 
of  the  beam,  whose  co-ordi- 
nates are  x  and  y,  the  pres- 

sures applied  to  AP,  and  in  equilibrium,  are  the  pressure 
P1  at  A,  the  load  px  supported  by  AP,  and  producing  the 
same  effect  as  though  it  were  collected  over  the  centre  of 
that  portion  of  the  beam,  and  the  elastic  forces  developed 
upon  the  transverse  section  of  the  beam  at  P  ;  whence  it 
follows  (Art.  360.)  by  the  principle  of  the  equality  of 
moments,  taking  P  as  the  point  from  which  the  moments 
are  measured,  that 


=W-I>  ....  (6*8). 

Integrating  this  equation  between  the  limits  a^  and  #,  and 
representing  the  inclination  to  the  horizon  of  the  tangent  to 
the  neutral  line  at  B  by  /32, 


El(-tan.  /3,)  = 


-<)  ....  (544). 


Integrating  again  between  the  limits  0  and  a?, 


.  (545), 


Whence  observing  that  when  x=al,  y= 
El  tan.  pa—  P2 


(546). 


Similarly  observing,  that  if  x  and  y  be  taken  to  represent 
the  co-ordinates  of  a  point  Q  in  the  beam  between  B  and  C, 
the  pressures  applied  to  AQ  are  the  elastic  forces  upon  the 
section  at  Q,  the  pressures  P,  and  Pa  and  the  load  ^x}  we 
have 


Integrating  this  equation  between  the  limits  al  and  a?,  and 

fl'tl 

observing  that  at  the  former  limit  the  value  of  -/-  is  repre- 
sented by  tan.  /3a,  we  have 


4:88  THE   DEFLEXION  OF   A   BEAM 

Big-tan.  0.)  =Hrf-«O-ff.('*-< 

....  (548). 

Now  it  is  evident  that,  since  the  props  B  and  C  are  placed 
symmetrically,  the  lowest  point  of  the  beam,  and  therefore 
of  the  neutral  line,  is  in  the  middle,  between  B  and  C  ;  so 

/m/M 

that  -/•  =  0,  when  x=a.     Making  this  substitution  in  equa- 

tion (548), 

-El  tan.  p^ipW-afi-tPW-a^-lY^a-aJ  .  .  (549). 

Since,  moreover,  the  resistances  at  C  and  D  are  equal  to 
those  at  B  and  A,  and  that  the  whole  load  upon  the  beam  is 
sustained  by  these  four  resistances,  we  have 

P.+P^f**  .....  (550). 

Assuming  ai=naj  and  eliminating  P1?  Pa,  tan.  ft,  between 
the  equations  (546),  (549),  and  (550),  we  obtain 

-D     pa  (  ™3-f-127ia—  24/1  +  8 


—  8 


24EI 


(5^a—  16^+8) 
'   1       2*-8    "f 


Making  a?=0  in  equation  (544);  and  observing  that  the  cor- 
responding value  of  -y-  is  represented  by  tan.  /31?  we  have 

El  (tan.  ft—  tan.  ft)==  —  |i*a1$+iP1a1i. 

Substituting  for  tan.  ^2  and  Px  their  values  from  equations 
(553)  and  (551),  and  reducing, 


) 
[....(554). 


Representing  the  greatest  deflexions  of  the  portions  AB  and 


LOADED   UNIFORMLY. 


489 


BC  of  the  beam,  respectively,  by  Dl  and  D2,  and  by  a?,  the 
distance  from  A  at  which  the  deflexion  Dj  is  attained,  we 
have,  by  equations  (544)  and  (545), 

-El  tan.  /3i== jf*  (x*-a*)-<, 
x— a^tan.  /32)=- 


The  value  of  Dj  is  determined  by  eliminating  a?,  between 
these  equations,  and  substituting  the  values  of  Pa  and  tan.  J3t 
from  equations  (551)  and  (553). 

Integrating  equation  (548)  between  the  limits  at  and  0, 
and  observing  that  at  the  latter  limit  y=D2,  we  have 


^EI^-^)  tan. 


Substituting  in  this  equation  for  the  values  of  tan.  )8S,  P,,  Pa, 
and  reducing,  we  obtain 


(556). 


Representing  BC  by  2#t,  and  observing   that   ^2  =  AE  — 
AB=a—  na=(l—  ri)a, 


^ 
-Ua~48EI  '    3-2ril-n 


375.    A   BEAM,    HAVING   A    UNIFORM   LOAD,   SUPPORTED   AT   EACH 
EXTREMITY,   AND   BY    A    SINGLE    STRUT   IN    THE   MIDDLE. 

If,  in  the  preceding  article,  a^  be  assumed  equal  to  a,  or 

w=l,  the  two  props  B  and 
C  will  coincide  in  the  centre ; 
and  the  pressure  P2  upon 
the  single  prop,  resulting 
from  their  coincidence,  win 
be  represented  by  twice  the 
corresponding  value  of  Pa  in 
equation  (552) ;  we  thus  ob- 
tain 


490 


THE  DEFLEXION    OF   A   BEAM 


The  distance  xl  of  the  point  of  greatest  deflexion  of  either 
portion  of  the  beam  from  its  extremities  A  or  D,  and  the 
amount  D,  of  that  greatest  deflexion,  are  determined  from 
equations.  (555).  Making  tan.  /32=0  in  those  equations, 
substituting  for  P,  its  value,  solving  the  former  in  respect  to 
0J,  and  the  latter  in  respect  to  D0  we  obtain 


16 


#=•4215350 (559). 


48EI 


48EI 


(560). 


376.    A   BEAM   WHICH    SUSTAINS   A   UNIFORM   LOAD   THROUGHOUT 
ITS   WHOLE   LENGTH.    AND   WHOSE  EXTREMITIES  ARE  SO  FIRMLY 


IMBEDDED 
RIGID. 


IN     A     SOLID     MASS     OF    MASONRY    AS     TO     BECOME 


Let  the  ratio  of  the  lengths  of  the  two  portions  AB  and 
AE  of  a  beam,  supported  by  two  props  (p.  487),  be  assumed 
to  be  such  as  will  satisfy  the  condition  5^2— 16?i  +  8=0 ;  or, 
solving  this  equation,  let 


n=- 


(561). 


The  value  of  tan.  £„ 


(equation  553)  will  then  become 
zero  ;  so  that  when  this  re- 
lation obtains,  the  neutral 
line  will,  at  the  point  B,  be 
parallel  to  the  axis  of  the 
abscissae  ;  or,  in  other  words, 
the  tangent  to  the  neutral 
line  at  the  point  B  will  retain, 
after  the  deflexion  of  the  beam,  the  position  which  it  had' 
before;  i.  e.,  its  position  will  be  that  which  it  would  have 
retained  if  the  beam  had  been,  at  that  point,  rigid.  Now 
this  condition  of  rigidity  is  precisely  that  which  results  from 
the  insertion  of  the  beam  at  its  extremities  in  a  mass  of 
masonry,  as  shown  in  the  accompanying  figure  ;  whence  it 
follows  that  the  deflexion  in  the  middle  of  the  beam  is  the 
same  in  the  two  cases.  Taking,  therefore,  the  negative  sign 
in  equation  (561),  and  substituting  for  n  its  value  f  (4—  1/6) 
or  -6202041  in  equation  (557),  and  observing  that,  in  that 


SUPPORTED   AT   ANY   NUMBER   OF   POINTS. 


491 


equation,  2#a  represents  the  distance  BC  in  the  accompany- 
ing figure,  we  obtain 


2±EI 


(562). 


By  a  comparison  of  this  equation  with  equation  (530),  it 
appears  that  the  deflexion  of  a  learn  sustaining  a  pressure 
uniformly  distributed  over  its  whole  length,  and  having  its 
extremities  prolonged  and  firmly  imbedded,  is  only  one-fifth 
of  that  which  it  would  exhibit  if  its  extremities  were  free.* 

If  the  masonry  which  rests  upon  each  inch  of  the  portion 
AB  of  the  beam  be  of  the  same  weight  as  that  which  rests 
upon  each  inch  of  BC,  the  depth  AB  of  the  insertion  of  each 
end  should  equal  '62  of  AE,  or  about  three  ten.hs  of  the 
whole  length  of  the  beam. 


1 

,  I1 

j 

f 

1 

,  J. 

4 

a          D 

4 

A 

J 

1 

i 

1 

I 

1 

1 

[ 

1 

—  r~ 

377.  Conditions  of  the  equilibrium  of  a  beam  supported  at 
any  number  of  points  and  deflected  by  given  pressures. 

To  simplify  the  investigation,  let  the  points  of  support 

ABC  be  supposed  to  be  three 
in  number,  and  let  the  direc- 
tions of  the  pressures  bisect 
the  distances  between  them ; 
the  same  analysis  which  de- 
termines the  conditions  of  the 
equilibrium  in  this  case  will 
be  found  applicable  in  the  more  general  case.  'Let  P15  P3, 
PB,  be  taken  to  represent  the  resistances  of  the  several  points 
of  support,  al  and  #2  the  distances  between  them,  PQ  P4  the 
deflecting  pressures,  and  x  y  the  co-ordinates  of  any  point  in 
the  neutral  line  from  the  origin  B.  Substituting  in  equation 

(500)  for  ^  its  value  -y^,  and  observing  that  in  respect  to  the 

portion  BD  of  the  beam  ^Pp=P9(^al— x)— 'P1(al— x\  and 
that  in  respect  to  the  portion  DA  of  the  beam,  2Pj9= 
— ]?,(#,— x\  we  have  for  the  differential  equation  to  the 
neutral  line  between  B  and  D 

*  The  following  experiment  was  made  by  Mr.  Hatcher  to  verify  this  result. 
A  strip  of  deal  y3^-  in.  byy7g-  in.  was  supported  with  its  extremities  resting 
loosely  on  rollers  six  feet  apart,  and  was  observed  to  deflect  1-2  inch  in  the 
middle  by  its  own  weight.  The  extremities  were  then  made  rigid  by  confining 
them  between  straight  edges,  and,  the  distance  between  the  points  of  support 
remaining  the  same,  the  deflexion  was  observed  to  be  '22  inch.  The  theory 
would  have  given  it  '24. 


492  BEAM   SUPPORTED   AT  AlfY  NUMBER  OF  POINTS. 


=P,(K-a!)-P,(«1-!B)  ----  (563), 
between  D  and  A 


^-P^-*)  .  .  .  (584). 

.Representing  by  £  the  inclination  of  the  tangent  at  B  to  the 
axis  of  the  abscissae,  and  integrating  the  former  of  these 
equations  twice  between  the  limits  0  and  #, 

ii.  ft  ....  (565); 


tan-  £  •  - 

Substituting  ^al  for  x  in  these  equations,  and  representing 
by  D!  the  value  of  y,  and  by  7  the  inclination  to  the  horizon 
of  the  tangent  at  the  point  D,  we  obtain 

El  tan.  y=£P2a12-fI)A2+EItan./3  ....  (567), 
Ero^APA'-APA'+iEL*,  tan.  0  .  .  .  .  (568). 
Integrating  equation  (564)  between  the  limits  -^  and  x 


tan. 

Eliminating  tan.  y  between  this  equation  and  equation  (567) 
and  reducing, 

EI^^-PX^-^+EI^-^+iPA2  •  •  •  (569). 

Integrating  again  between  the  limits  -±  and  a?,  and   elimi- 
nating the  value  of  D,  from  equation  (568), 


]S"ow  it  is  evident  that  the  equation  to  the  neutral  line  in 
respect  to  the  portion  CE  of  the  beam,  will  be  determined 
by  writing  in  the  above  equation  P5  and  P4  for  Px  and  P, 
respectively. 

Making  this  substitution  in  equation  (570),  and  writing 
-tan.  /3  for  +tan.  (3  in  the  resulting  equation  ;  then  assum- 


BEAM   SUPPORTED   AT   ANY   NUMBER   OF   POINTS.  493 

ing  x=a1  in  equation  (570),  and  a?=#a  in  the  equation  thus 
derived  from  it,  and  observing  that  y  then  becomes  zero  in 
both,  we  obtain 


0=-iPA'  +  s  PA'+EI^  tan.  ft 
A8-EIa2  tan.  (3. 


Also,  by  the  general  conditions  of  the  equilibrium  of  parallel 
pressures  (Art.  15.), 


Eliminating  between  these  equations  and  the  preceding,  as- 
suming a1  +  a.t=a^  and  reducing,  we  obtain  . 


1600, 


By  equation  (568), 

D.=7 

Similarly, 


-9I>A1    .....  (5T5); 


'By  equation  (567), 

.....  (577). 


If  a,  be  substituted  for  x  in  equation  (569),  and  for  P,  and 
tan.  /3  their  values  from  equations  (571)  and  (576)  ;  and  if 
the  inclination  of  the  tangent  at  A  to  the  axis  of  x  be  repre- 
sented by  p,j  we  shall  obtain  by  reduction 


494:  BEAM   SUPPORTED   AT   ANT   NUMBER    OF   POINTS. 


.....  (578). 

Similarly,  if  /32  represent  the  inclination  of  the  tangent  at 
C  to  the  axis  of  a?, 

.....  (579). 


378.  If  the  pressures  P2  and  P4,  and  also  the  distances  0, 
tnd  aa,  be  equal, 

P,=P5=AP5,  P,=  VP«tan.  0=0,  tan.  ft=tan.  0.= 


379.  If  the  distances  ^  and  a2  be  equal,  and  P4=3Pa, 
P,=iP,,  P.=VP«  P.=tP0  tan.  /3=-^fi,  tan./3,=0.* 

380.  If  «,=»,  and  3P,=13Pa,  P,=0,  P  ^  ',  P,,.P5=fP,. 

*  The  following  experiments  were  made  by  Mr.  Hatcher  to  verify  this  result. 
The  bar  ACB,  on  which  the  experiment  was  to  be  tried,  was  supported  on 
knife  edges  of  wrought  iron  at  A,  C,  and  B,  whose  distances  AC  and  CB  were 
each  five  feet.  The  angles  of  the  knife  edges  were  90°,  and  the  edges  were 
oled  previous  to  the  experiments.  The  weights  were  suspended  at  points  D 


and  E  intermediate  between  the  points  of  support.  In  measuring  the  angles 
of  deflexion  the  instrument  (which  was  a  common  weighted  index-hand  turn- 
ing on  a  centre  in  front  of  a  graduated  arc)  was  placed  so  that  the  angle  c 
of  vhe  parallelogram  of  wood  carrying  the  arc  was  just  over  the  knife-edge  B, 
the  side  cd  of  the  parallelogram  resting  on  the  deflected  bar.  This  position 
gave  the  angle  at  the  point  of  support. 

1st  Experiment. — A  bar  of  wrought  iron  half  an  inch  square,  being  loaded 
at  E  with  a  weight  of  18  Ib.  13  oz.,  and  at  D  with  52  Ib.  3  oz.,  assumed  a  per- 
fectly horizontal  position  at  B,  as  shown  by  the  needle.  The  proportion  of 
these  weights  is  2-77  : 1. 

2d  Experiment. — A  bar  -7  inches  square,  being  loaded  at  E  with  a  weight  of 
87 '3  Ib.,  and  at  D  with  a  weight  of  112  Ib.,  assumed  a  perfectly  horizontal 
position  at  B.  The  weights  were  in  this  experiment  accurately  in  the  propor- 
tion 3 :  1. 

3d  Experiment. — A  round  bar,  '75  inch  in  diameter,  being  loaded  at  E  with 
37 '3  Ib.,  and  at  D  with  112  Ib.,  showed  a  deviation  from  the  horizontal  position 
at  B  amounting  to  not  more  than  20'.  The  weights  were  in  the  proportion  of 
8:1. 

The  influence  of  the  weight  of  the  bar  is  not  taken  into  account. 


A   BEAM   DEFLECTED   BY    PRESSURES. 


495 


381.  CURVATURE  OF  A  RECTANGULAR  BEAM,  THE  DIRECTION  OF 
THE  DEFLECTING  PRESSURE  AND  THE  AMOUNT  OF  THE  DE- 
FLEXION BEING  ANY  WHATEVER. 

The  moment  of  inertia  I  (Art.  358.)  is  to  be  taken,  about 
an  axis  perpendicular  to  the  plane  of  deflexion,  and  passing 
through  the  neutral  line,  the  distance  h  of  which  neutral 
line  from  the  centre  of  gravity  of  the  section  is  determined 
by  equation  (499). 

Now  y^fo3  representing  (Art.  362.)  the  moment  of  inertia 
of  the  rectangular  section  of  the  beam  about  an  axis  pass- 
ing through  its  centre  of  gravity,  it  follows  (Art.  79.)  that 
the  moment  I  about  an  axis  parallel  to  this  passing  through 
a  point  at  distance  h  from  it  is  represented  by 


Substituting,    therefore,    the    value    of    h    from    equation 
(499), 


Substituting  this  value  in  equation  (500),  and  reducing, 
1_         12P,E^ 
R~~12RfP11sin.1d  +  EW  ' 

Draw  ax  parallel  to  the 
position  of  the  beam  be- 
fore deflexion;  take  this 
line  as  the  axis  of  the 
abscissae  and  a  as  the 
origin  ;  then  pl  =Jlm=It,n 

+nm=MJR   cos. 

aM.  sin. 

-\-x  sin.  M&m. 

Let,  now,  the  inclination 
DaPl  of  the  direction  of 
P!  to  the  normal  at  a  be 
represented  by  d,,  and  the  inclination  Mat  of  the  tangent  to 

the  neutral  line  at  a  to  a%,  by  f31  ;  then 


^    cos- 


:.p,=y  sn.     1+/1-f»  cos. 
Substituting  this  value  of  pl  in  the  preceding  equation, 


496          A  BEAM  DEFLECTED  BY  PRESSURES. 


.  in.  ft  +ft)+a  cos.  (*.+£,){  ._.. 

K~  12RT2sin.2^+E^V 


where  0  represents  (Art.  355.)  the  inclination  Rqa  of  the 
normal  at  the  point  K  to  the  direction  of  P,. 


382.  Case  in  which  the  deflexion  of  the  beam  is  small. 

If  the  deflexion  be  small,  and  the  inclination  d,,  of  the 
direction  of  Pt  to  the  normal  at  its  point  of  application,  be 


not  greater  than  j  ;  then  y  sin.  (^+/3a)  is  exceedingly  small, 


and  may  be  neglected  as  compared  with  x  cos.  (^1+/31);  in 
this  case,  moreover,  6  is,  for  all  positions  of  R,  very  nearly 
equal  to  6im  Neglecting,  therefore,  £,  as  exceedingly  small, 
we  have 


!._  .. 

R~12RT12  sin.  f^+E'JV  ' 


Solving  this  equation,  of  two  dimensions,  in  respect  to  =p  ,  and 
taking  the  greater  root, 
1      6Pt 


.a^—  icasin.2^}  ----  (584). 


383.  THE  WOKE:  EXPENDED  UPON  THE  DEFLEXION  OF  A  UNI- 

FORM RECTANGULAR  BEAM,  WHEN  THE  DEFLECTING  PRES- 
SURES ARE  INCLINED  AT  ANY  ANGLE  GREATER  THAN  HALF  A 
RIGHT  ANGLE  TO  THE  SURFACE  OF  THE  BEAM. 

If  u^  represent  work  expended  on  the  deflexion  of  the 
portion  AM  of  the  beam,  then  (equation  505) 


but  by  equation  (500)%s=p  .  -g 


A:   BEAM   DEFLECTED   BY    PRESSURES.  497 


p,      6P 

~ 


by  equation  (584),  observing  that  the  deflexion  being  small, 
p^—x  cos.  6l  very  nearly.  Now  the  value  of  ^  (equation 
584)  becomes  impossible  at  the  point  where  a?  cos.  ^becomes 

less  than — —c  sin.  &.  ;  the  curvature  of  the  neutral  line  com- 

1/3 

mences  therefore  at  that  point,  according  to  the  hypotheses; 
on  which  that  equation  is  founded.  Assuming,  then,  the 

corresponding  value  —7=0  tan.  d,  of  a?  to  be  represented  by  xt> 

the  integral  (equation  585)  must  be  taken  between  the  limits 
a?,  and  a»  instead  of  0  and  at ; 

SP/cosX/!  •  -7-i *z — i  i  •     ** )  j 

/.  uj= — ^j—9 — /  \x  cos.  O.+x  Vx  cos.  «L— wr  sin.  ",J  o»  J 
J^y(?    */ 

«i 

P a  cos  2<M         1  3 1 

/,^=    x        '    H^3— -       C3tan.3t).  +  (^a— ^atan.a^)2  h*(586). 

xLOC  l  ,-i  >4/o  i/     j     \         / 


And  a  similar  expression  being  evidently  obtained  for  the 
work  expended  in  the  deflexion  of  the  portion  BM  of  the 
beam,  it  follows,  neglecting  the  term  involving  o3  as  exceed- 
ingly small  when  compared  with  &,3,  that  the  whole  work  TJ^ 
expended  upon  the  deflexion  is  represented  by  the  equation 

°  C08'  '*  i-      tan.          + 


But  if  ^3  be  taken  to  represent  the  inclination  of  P3  to  the- 
normal  to  the  surface  of  the  beam,  as  6l  and  da  represent  the 
similar  inclinations  of  Pl  and  P2,  then,  the  deflexion  being 
small, 

*  Church's  Int.  Cal.     Art.  149. 
32 


498          A  BEAM  DEFLECTED  BY  PRESSURES. 


^  COS.      =,aa  COS.    ,,       2&  COS.    2  =     3«1  COS.    ,. 

Eliminating  Pt  and  P2  between   these   equations  and  the 
preceding, 


(587). 

If  the  pressure  Ps  be  applied  perpendicularly  in  the  centre 
of  the  beam,  and  the  pressures  Px  and  P2  be  applied  at  its 
extremities  in  directions  equally  inclined  to  its  surface  ;  then 
^=(1^=^^  6l=6t=6^  and  dg=0.  Substituting  these  values 
in  the  preceding  equations,  and  reducing, 

....  (588). 


384.  THE  LINEAR  DEFLEXION  OF  A  RECTANGULAR  BEAM. 

Dj  being  taken  as  before  (Art.  368.)  to  represent  the  de- 
flexion of  the  extremity  A  measured  in  a  direction  perpen- 
dicular to  the  surface  of  the  beam,  we  have  (Art.  52.) 


^I  cos. 


But  by  equation  (586),  neglecting  the  term  involving  c8, 

CC(/Cv1  -^  j_     ,  QAf  ^./  Q  10,  O*\"Oi 

a2 


Dividing  both  sides  by  P15  reducing,  and  integrating, 

9P  2. 

D.^^cos^.j^  +  K-ic'tan.  'O,)2}  ----  (589) 

Proceeding  similarly  in  respect  to  the  deflection  D8  perpen 


INCLINED   AT   ANY   ANGLE   TO    ITS    SURFACE.  499 

dicnlar  to  the  surface  of  the  beam  at  the  point  of  application 
of  P3,  we  obtain  from  equation  (587) 


.  .  .  .  (590) 


In  the  case  in  which  P,  and  P3  are  equally  inclined  to  the 
extremities  of  the  beam  and  the  direction  of  P3  bisects  it, 
this  equation  becomes 


385.  The  work  expended  upon  the  deflexion  of  a  beam  sub- 
jected £p  the  action  of  pressures  applied  to  its  extremities, 
and  to  a  single  intervening  point,  and  also  to  the  action 
of  a  system  of  parallel  pressures  uniformly  distributed 
over  its  length. 

Let  «-  represent  the  aggregate  amount  of   the  parallel 


pressures  distributed  over  each  unit  of  the  length  of  the 
beam,  and  a  their  common  inclination  to  the  perpendicular 
to  the  surface  ;  then  will  px  represent  the  aggregate  of  those 
distributed  uniformly  over  the  surface  DT,  and  these  will 
manifestly  produce  the  same  effect  as  though  they  were 
collected  in  the  centre  of  DT.  Their  moment  about  the 
point  R  is  therefore  represented  by  M-OJJOJ  cos.  a,  or  by  ^x3 
cos.  a  ;  and  the  sum  of  the  moments  of  the  pressures  applied 
to  AT  is  represented  by  (P^  cos.  da—  Jjxaj*  cos.  a).  Substi- 
tuting this  value  of  the  sum  of  the  moments  for  Pj?a  in 
equation  (505),  we  obtain 


_  1      /*(?,#  COS.  tj—jwi?  COS.  a)a 

'  ~~  ~ 


500 


DEFLEXION    OF   A   BEAM   BY   PRESSURES. 


386.  If  the  pressures  le  all  perpendicular  to  the  surface  of 
the  ~beam,  ^=0,  a=0,  and  I  is  constant  (equation  499); 
whence  we  obtain,  by  integration  and  reduction, 


(592). 


If  the  pressure  P3  be  applied  in  the  centre  of  the  beam, 
P^-JP.+i^fl,  and  a1=^aj  also  the  whole  work  U3  of 
deflecting  the  beam  is  equal  to  2^;  whence,  substituting 
and  reducing, 


(593)- 


387.  A  RECTANGULAR  BEAM  IS  SUPPORTED  AT  ITS  EXTREMITIES 
BY  TWO  FIXED  SURFACES,  AND  LOADED  IN  THE  MIDDLE  I  IT 
IS  REQUIRED  TO  DETERMINE  THE  DEFLEXION,  THE  FRICTION 
OF  THE  SURFACES  ON  WHICH  THE  EXTREMITD3S  REST  BEING 
TAKEN  INTO  ACCOUNT. 

It  is  evident  that  the  work  which  produces  the  deflexion 


of  the  beam  is  done  upon  it  partly  by  the  deflecting  pressure 
P,  and  partly  by  the  friction  of  the  surface  of  the  beam 
upon  the  fixed  points  A  and  B,  over  which  it  moves  whilst 
in  the  act  of  deflecting.  Representing  by  9  the  limiting 
angle  of  resistance  between  the  surface  of  the  beam  and 
either  of  the  surfaces  upon  which  its  extremity  rests,  the 
friction  Q,  or  Q2  upon  either  extremity  will  be  represented 
by  fP  tan.  9 ;.  and  representing  by  s  the  length  of  the 
curve  ca  or  cb,  and  by  %a  the  horizontal  distance  between 
the  points  of  support ;  the  space  through  which  the  surface 
of  the  beam  would  have  moved  over  each  of  its  points  of 
support,  if  the  point  of  support  had  been  in  the  neutral  line, 
is  represented  by  s— a,  and  therefore  the  whole  work  done 
upon  the  beam  by  the  friction  of  each  point  of  support  by 

i  tan.  <p/P<&.     Moreover,.  I>  representing  the  deflexion  of 


THE   SOLID   OF   THE    STRONGEST   FORM.  501 

the  beam  under  any  pressure  P,  the  whole  work  done  by  P 
is  represented  by  /P^D.  Substituting,  therefore,  for  the 

work  expended  upon  the  elastic  forces  opposed  to  the 
deflexion  of  the  beam  its  value  from  equation  (588),  and  ob- 
serving that  the  directions  of  the  resistances  at  A  and  B  are 
inclined  to  the  normals  at  those  points  at  angles  equal  to 
the  limiting  angle  of  resistance,  we  have 


/ 


But 


e(luati011 

Substituting  these  values  in  the  above  equation,  and  dif- 
ferentiating in  respect  to  P,  we  have 

T>dD_P  K  +  02—|ca  tan.  a<p)f|       PV 

"•  ~30ETtan'  9' 


Dividing  by  P,  and  integrating  in  respect  to  P, 

PK  +  (^-^tan.^       PV 

3  "60ET        9  ' 


388.  THE   SOLID    OF    THE    STRONGEST   FORM   WITH   A    GIVEN 
QUANTITY  OF  MATERIAL. 

The  strongest  form  which  can  be  given  to  a  solid  body  in 
the  formation  of  which  a  given  quantity  of  material  is  to  be 
used,  and  to  which  the  strain  is  to  be  applied  under  given 
circumstances,  is  that  form  which  renders  it  equally  liable  to 
rupture  at  every  point.  So  that  when,  by  increasing  the 
strain  to  its  utmost  limit,  the  solid  is  brought  into  the  state 
bordering  upon  rupture  at  one  point,  it  may  be  in  the  state 
bordering  upon  rupture  at  every  other  point.  For  let  it  be 
supposed  to  be  constructed  of  any  other  form,  so  that  its 
rupture  may  be  about  to  take  place  at  one  point  when  it  is 
not  about  to  take  place  at  another  point,  then  may  a  portion 
of  the  material  evidently  be  removed  from  the  second  point 
without  placing  the  solid  there  in  the  state  bordering  upon 


502  THE   RUPTURE    OF   A   BAR. 

rupture,  and  added  at  the  first  point,  so  as  to  take  it  out 
of  the  state  bordering  upon  rupture  at  that  point ;  and  thus 
the  solid  being  no  longer  in  the  state  bordering  upon 
rupture  at  any  point,  may  be  made  to  bear  a  strain  greater 
than  that  which  was  before  upon  the  point  of  breaking  it, 
and  will  have  been  rendered  stronger  than  it  was  before. 
The  first  form  was  not  therefore  the  strongest  form  of  which 
it  could  have  been  constructed  with  the  given  quantity  of 
material;  nor  is  any  form  the  strongest  which  does  not 
satisfy  the  condition  of  an  equal  liability  to  rupture  at  every 
point. 

The  solid,  constructed  of  the  strongest  form,  with  a  given 
quantity  of  a  given  material,  so  as  to  be  of  a  given  strength 
under  a  given  strain,  is  evidently  that  which  can  be  con- 
structed, of  the  same  strength,  with  the  least  material ;  so 
that  the  strongest  form  is  also  the  form  of  the  greatest 
economy  of  material. 

RUPTURE. 

389.  The  rupture  of  a  bar  of  wood  or  metal  may  take 
place  either  by  a  strain  or  tension  in  the  direction  of  its 
length,  to  which  is  opposed  its  TENACITY  ;  or  by  a  thrust  or 
compressing  force  in  the  direction  of  its  length,  to  which  is 
opposed  its  resistance  to  COMPRESSION  ;  or  each  of  these 
forces  of  resistance  may  oppose  themselves  to  its  rupture 
transversely,  the  one  being  called  into  operation  on  one  side 
of  it,  and  the  other  on  the  other  side,  as  in  the  case  of 
a  TRANSVERSE  STRAIN. 


TENACITY. 

390.  The  tenacities  of  different  materials  as  they  have 
been  determined  by  the  best  authorities,  and  by  the  mean 
results  of  numerous  experiments,  will  be  found  stated  in  a 
table  at  the  end  of  this  volume.  The  unit  of  tenacity  is  that 
opposed  to  the  tearing  asunder  of  a  bar  one  square  inch  in 
section,  and  is  estimated  in  pounds.  It  is  evident  that  the 
tenacity  of  a  fascile  of  n  such  bars  placed  side  by  side,  or 
of  a  single  bar  n  square  inches  in  section,  would  be  equal 
to  n  such  units,  or  to  n  times  the  tenacity  of  one  bar. 

To  find,  therefore,  the  tenacity  of  a  bar  of  any  material 
in  pounds,  multiply  the  number 'of  square  inches  in  its  sec- 


RUPTURE   OF   A   BAR    SUSPENDED   VERTICALLY.  503 

tion   by  its  tenacity  per  square    inch,  as   shown  by  the 
table.  ' 


391.    A   BAR,    CORD,  OR   CHAIN   IS    SUSPENDED   VERTICALLY,  CAR- 
RYING  A   WEIGHT     AT   ITS   EXTREMITY  I     TO     DETERMINE     THE 

'CONDITIONS  OF  ITS  RUPTURE. 

First.  Let  the  bar  be  conceived  to  have  a  uniform  section 
represented  in  square  inches  by  K  ;  let  its  length  in  inches 
be  L,  the  weight  of  each  cubic  inch  |u<,  the  weight  suspended 
from  its  extremity  W,  the  tenacity  of  its  material  per  square 
inch  r  ;  and  let  it  be  supposed  capable  of  bearing  m  times 
the  strain  to  which  it  is  subjected.  The  weight  of  the  bar 


will  then  be  represented  by  juJLK,  and  the  strain  upon  its 
highest  section  by  ^LK-h  W.  Now  the  strain  on  this  section 
is  evidently  greater  than  that  on  any  other  ;  it  is  therefore  at 
this  section  that  the  rupture  will  take  place.  But  the  resist- 
ance opposed  to  its  rupture  is  represented  by  Kr  ;  whence  it 
follows  (since  this  resistance  is  m  times  the  strain)  that 


(595). 


By  which  equation  is  determined  the  uniform  section  K  of  a 
bar,  cord,  or  chain,  so  that  being  of  a  given  length  it  may  be 
capable  of  bearing  a  strain  m  times  greater  than  that  to 
which  it  is  actually  subjected  when  suspended  vertically. 
The  weight  Wj.  of  the  bar  is  represented  by  the  formula 


392.  Secondly.  Let  the  section  of  the  rod  be  variable  ;  and 
let  this  variation  of  the  section  be  such  that  its  strength,  at 
every  point,  may  be  that  which  would  cause  it  to  bear, 
without  breaking,  m  times  as  great  a  strain  as  that  which  it 
actually  bears  there.  Let  K  represent  this  section  at  a  point 
whose  distance  from  the  extremity  which  carries  the  weight 
W  is  x  ;  then  will  the  weight  of  the  rod  beneath  that  point 

be  represented  by  I  &~Kdx  ;  or,  supposing  the  specific  gravity 


504  RUPTURE   OF  A   BAR   SUSPENDED   VERTICAL!  ,T. 


of  the  material  to  be  every  where  the  same,  by  f/Kdfo?  :  also 
the  resistance  of  this  section  to  rupture  is  Kr. 


Differentiating  this  expression  in  respect  to  a?,  observing  that 
K  is  a  function  of  a?,  and  dividing  by  Kr,  we  obtain 

1  dK  _  mp 
K  ^~  =  :T; 

Integrating  this  expression  between  the  limits  0  and  #,  and 
representing  by  Ko  the  area  of  the  lowest  section  of  the  rod, 


:But  the  strain  sustained  by  the  section  KO  is  W,  therefore 


(597). 


The  whole  weight  W2  of  the  rod,  cord,  or  chain,  is  repre- 
sented by  the  formula 


-l     .  •  •    (598). 


A  rope  or  chain,  constructed  according  to  these  conditions, 
is  evidently  as  strong  as  the  rope  or  chain  of  uniform  section 
whose  weight  TV\  is  determined  by  equation  (596),  the  value 
of  77i  being  taken  the  same  in  both  cases.  The  saving  of  ma- 
terial effected  by  giving  to  the  cord  or  chain  a  section  vary- 
ing according  to  the  law  determined  by  equation  (598)  is 
represented  by  "Wj — W2,  or  by  the  formula 


T^-T  i^rriL       v 

™^_W   ~~1     (599). 

r—m^L          V*  / 


*  Church's  Int.  Cal.     Art.  159. 


THE   SUSPENSION   BKIDGE. 


505 


THE  SUSPENSION  BRIDGE. 

393.  General  conditions  of  the  equilibrium  of  a  loaded 

chain. 

Let  AEH  represent  a  chain  or  cord  hanging  freely  from 

two  fixed  points  A  and  H, 
and  having  certain  weights 
w. 


\ 


f  •  I  T  7 .-  *  t 


rods 


M  wn  &c.,  suspended  by 
or    cords    from    given 
points  B,  C,  D,  &c.,  in  its 
length.     Through  the  lowest 
point  E  of  the  chain  draw 
the  vertical  E#,   containing 
as  many  equal  parts  as  there 
are  units  in   the  weight  of 
the  chain  between  E  and  any 
point   of   suspension   B,   to- 
gether with  the  suspending 
rods  attached  to  it,  and  the  weights  which  they  severally 
carry ;  draw  aP  parallel  to  the  direction  of  a  tangent  to  the 
curve  at  B,  and  produce  the  tangent  at  E  to  meet  aP  in  P ; 
then  will  aP  and  EP  contain  as  many  equal  parts  as  there 
are  units  in  the  tensions  at  B  and  E  respectively  ;  and  if  E& 
and  EC  be  taken  to  represent  the  whole  weights  sustained  by 
EC  and  ED,  and  P5  and  PC  be  joined,  these  lines  will  in 
like  manner  represent  the  tensions  upon  the  points  C  and  D. 
For  the  pressures  applied  to  EB,  and  in  equilibrium,  being 
the  weight  of  the  chain,  the  weights  of  the  suspending  rods, 
the  weights  attached  to  the  rods,  and  the  tensions  upon  B 
and  E,  the  principle  of  the  polygon  of  pressures  (Art.  9.) 
obtains  in  respect  to  these  pressures.     Now  the  lines  drawn 
to   complete   this   polygon,  parallel   to   the   weights,  form 
together  the  vertical  line  E&,  and  the  polygon  (resolving 
itself  into  a  triangle)  is  completed  by  the  lines  aP  and  EP 
drawn  parallel  to  the  tensions  upon  B  and  E.     Each  line 
contains,  therefore,  as  many  equal  parts  (A^.  9.)  as  there 
are  units  in  the  corresponding  tension.     Also,  the  pressures 
applied  to  the  portion  EC  of  the  curve,  being  the  weights 
whose  aggregate  is  represented  by  E&,  and  the  tensions  upon 
E  and  C,  of  which  the  former  is  represented  in  direction 
and  amount  by  EP,  it  follows  (Art.  9.)  that  the  latter  is 
represented  also  in  direction  and  amount  by  the  line  P5, 


506 


THE   CATENARY. 


which  completes  the  triangle  aPb ;  so  that  5P  is  parallel  to 
the  tangent  at  C. 

In  like  manner  it  is  evident  that  the  tension  upon  D  is 
represented  in  magnitude  and  direction  by  cP ;  so  that  cP  is 
parallel  to  the  tangent  to  the  curve  at  D. 


THE   CATENARY. 

394.  If  a  chain  of  uniform  section  be  suspended  freely 
between  two  fixed  points  A  and  B,  being  acted  upon  by  no 
other  pressures  than  the  weights  of  its  parts,  then  it  will 
assume  the  geometrical  form  of  a  curve  called  the 


Let  PT  be  a  tangent  to  any  point  P  of  the  curve  inter- 
secting the  vertical  CD  passing  through  its  lowest  point  D 


in  T ;  draw  the  horizontal  line  DM  intersecting  PT  in  Q ; 
take  this  line  as  the  axis  of  the  abscissae ;  and  let  DM  =a?, 
MP=y,  DP =5,  weight  of  each  unit  in  the  length  of  the 
chain  =f*,  tension  at  D— <?.  Now  DT  being  taken  to  repre- 
sent the  weight  w  of  DP,  it  has  been  shown  (Art.  393.) 
that  DQ  will  represent  the  tension  c  at  D,  and  TQ  that 
at  P. 

/7v/  DT     «.« 

Also,  jg=  tan.  PQM  =  tan.  DQT  =  j)Q=y , 


" dx 


(600). 


dx 


be- 


THE   CATENARY.  507 

tween  the  limits  0  and  s,*  and  observing  that  when  $=0, 


By  addition  and  reduction, 

i*x 

.....  (602> 


/  ^     —  i*x\ 

=*?(/_.") 


Substituting  this  value  for  s  in  equation  (600),  and  inte- 
grating between  the  limits  0  and  a?, 


~~ 


+s         _2  f*s    _s 


which  is  the  equation  to  the  catenary. 


395.  The  tension  (c)  on  the  lowest  point  of  the  catenary. 

Let  2S  represent  the  whole  length  of  the  chain,  and  2a 
the  horizontal  distance  between  the  points  of  attachment. 
Now  when  x=a,  s=S  ;  therefore  (equation  602), 

/  f*a       —  f*a\ 

s=£|  T    ~l (so*); 


for  which  expression  the  value  of  c  may  be  determined  by 
approximation. 


396.  The  tension  at  any  point  of  the  chain. 
The  tension  T  at  P  is  represented  by  TQ= 


Church's  Int.  Cal.     Art.  144. 


508  THE   CATENARY. 


(605). 


Now  the  value  of  c  has  heen  determined  in  the  preceding 
article  ;  the  tension  upon  any  point  of  the  chain  whose  dis- 
tance from  its  lowest  point  is  s  is  therefore  known. 


397.    The  inclination  of  the  curve  to  the  vertical  at  any 

point. 

Cl ?/ 

Let  i  represent  this  inclination,  then  cot.  i=-jr  • 

(  —      ~^x\ 
c         c     ] (606). 
£     -£        / 

The  inclination  may  be  determined  without  having  first 
determined  the  value  of  <?,  by  substituting  cot.  i  for  —  in 

equation  (601) ;  we  thus  obtain,  writing  also  a  and  S  for  x 
and  #, 


a 


q=tan.  i  log.  (cot.  i  +  cosec.  *)=tan.  i  log.     cot.  Jt; 

a 
8 


/.  —tan.  i  log.    tan.  j^s== (607). 


This  equation  may  readily  be  solved  by  approximation ;  a 
the  value  of  c  may  then  be  determined  by  the  equati 


and 
equation 


=^    tan.  i. 


398.  A.  chain  of  given  length  being  suspended  between  two 
given  points  in  the  same  horizontal  line :  to  determine  the 
depth  of  the  lowest  point  beneath  the  points  of  attachment  y 
and,  conversely,  to  determine  the  length  of  the  chain  whose 
lowest  point  shall  hang  at  a  given  depth  below  its  points 
of  attachment. 

The  same  notation  being  taken  as  before, 


THE   CATENARY.  509 

Integrating  between  the  limits  0  and  s,  and  observing  that 
y=0  when  s=0, 


Solving  this  equation  in  respect  to  *, 


+  j  .....  (609). 

If  H  represent  the  depth  of  the  lowest  point,  or  the  versed 
sine  of  the  curve,  then  y=H  when  «=S. 


(610). 


- (611). 


399.  The  centre  of  gravity  of  the  catena/ry. 

If  Gr  represent  the  height  of  the  centre  of  gravity  above 
the  lowest  point,  we  have  (Art.  32.) 


S.G=fyds=fy~dx. 

erefore,  for  y  am 
tions  (602)  and  (603),  we  have    : 


Substituting,  therefore,  for  y  and  — -  their  values  from  equa- 

cux 


L. 

— fix 


+s         +2  —  2  ^ 


510 


THE    SUSPENSION   BRIDGE 


But  by  equation  (604)  B=^-l  c        c    )  and   by  equation 

V   'g      — S         I 

603), 

-p 

s       — 4' 


+  ' 


.-.   0=iH-l- (613). 


400.   THE   SUSPENSION   BEIDGE   OF   GREATEST   STRENGTH,   THE 
WEIGHT  OF  THE  SUSPENDING  RODS  BEING  NEGLECTED. 

Let  ADB  represent  the  chain,  EF  the  road-way ;  and  let 


the  weight  of  a  bar  of  the  material  of  the  chain,  one  square 
inch  in  section  and  one  foot  long,  be  represented  by  f*,,  the 
weight  of  each  foot  in  the  length  of  the  road-way  by  f*a,  the 
aggregate  section  of  the  chains  at  any  point  P  (in  square 
inches)  by  K,  the  co-ordinates  DM  and  MP  of  P  by  a?  and  y, 
and  the  length  of  the  portion  DP  of  the  chain  by  s.  Then 

will  the  weight  of  DP  be  represented  by  ^  I  K<&,  and  the 

weight  of  the  portion  CM  of  the  roadway  by  ^x ;  so  that 
the  whole  load  (u)  borne  by  the  portion  DP  of  the  chain 
will  be  represented  (neglecting  the  weight  of  the  suspending 
rods)  by 

(614). 


OF   GREATEST    STRENGTH.  511 

Let  this  load  (w),  supported  by  the  portion  DP  of  the 
chain,  be  represented  by  the  line  D#,  and  draw  Dp  in  the 
direction  of  a  tangent  at  D,  representing  on  the  same  scale 
the  tension  c  at  that  point  ;  then  will  ap  be  parallel  to  a 
tangent  to  the  chain  at  P  (Art.  393). 


.. 

dx 


(615). 


Now  let  it  be  assumed  that  the  aggregate  section  of  the 
chains  is  made  so  to  vary  its  dimensions,  that  their  strength 
may  at  every  point  be  equal  to  m  times  the  strain  which 
they  have  there  to  sustain.  But  this  strain  is  represented  in 
magnitude  by  the  line  ap  (Art.  393.),  or  by  (ca-fV)*;  if, 
therefore,  r  be  taken  to  represent  the  tenacity  of  the  mate- 
rial of  the  chain,  per  square  inch  of  the  section,  then 


(616). 
Therefore  Kr=mc(l  +  ~j  =  me  (l  +  ^J*  (equation  615) 


=mc-j-  ;  therefore-/——1".     Also  /  ~Kds=  /K-^- 
dx  dx    mo  J  J      dx 

(equation  616)  ; 
/.(equation  6I4:)u=-2 


Differentiating  in  respect  to  a?,  and  observing  that  -r- 
du  dy    u  du  . 

Ty  a*r-0  Ty  (eiuatlon  615). 

du    u  du 


u  u 

re    r         du  r     T        udu 


512  THE    SUSPENSION   BRIDGE 

Integrating  these  expressions,*  we  obtain 


+ 


Substituting  in  this  equation  the  value  of  u  given  by  the 
preceding  equation,  and  reducing, 


which  is  the  equation  to  the  suspension  chain  of  uniform 
strength,  and  therefore  OF  THE  GREATEST  STRENGTH  WITH  A 

GIVEN   QUANTITY    OF   MATERIAL. 


401.  To  determine  the  variation  of  the  section  K  of  the 
chain  of  the  suspension  bridge  of  the  greatest  strength. 

Let  the  value  of  u  determined  by  equation  (617)  be  sub- 
stituted in  equation  (616)  ;  we  shall  thus  obtain  by  reduction 


(l  +  ^-)*«  }*  .  (819).t 
r    \         cmpj 


It  is  evident  from  this  expression  that  the  area  of  the  sec- 
tion of  the  chains,  of  the  suspension  bridge  of  uniform 
strength,  and  therefore  of  the  greatest  economy  of  material^ 
increases  from  the  lowest  point  towards  the  points  of  suspen- 
sion, where  it  is  greatest. 


*  Church's  Int.  Cal.     Art.  133,  Case  IV. 

•f  —  =  —  ;  .-.  s=  —  /Kdx.     Now  the  function  K  (equation  619)  may  be 
1   dx     me'  mcj 

integrated  in  respect  to  x  by  known  rules  of  the  integral  calculus  ,  the  value 
of  s  may  therefore  be  determined  in  terms  of  #,  and  thence  the  length  in 
terms  of  the  span.     The  formula  is  omitted  by  reason  of  its  length. 
Church's  Int.  Cal.     Art.  1  29,  Case  II. 


OF    GREATEST    STBKNGTH. 


513 


402.  To  determine  the  weight  "W  of  the  chain  of  the  suspen- 
sion bridge  of  the  greatest  strength. 

Let  it  be  observed  that  W=f*1  /  ~K.ds=u—  ^x    (equation 

614)  ;  substituting  the  value  of  u  from  equation  (617),  we 
have 


^-'-)*tan.  I  ^(l+-TA\^  I  -w  .     .  (620). 
mwj  (    r    \       cmpj      (  v      ' 


403.  To  determine  the  tension  c  upon  the  lowest  point  D  of' 
the  chain  of  uniform  strength. 

Let  H  be  taken  to  represent  the  depth  of  the  lowest  point 
D,  beneath  the  points  of  suspension,  and  2&  the  horizontal 
distance  of  those  points  :  and  let  it  be  observed  that  H  and 
a  are  corresponding,  values  of  y  and  x  (equation  618)  ; 

TT  T      1  (    W1, 

,VH= log.   sec.  \  -—± 

m^  (    r 

Solving  this  equation  in  respect  to  <?, 


c=- 


-1  ™ 


sec. 


-1 


..(621). 


404.    THE   SUSPENSION  BRIDGE   OF  GREATEST   STRENGTH,  THE 
WEIGHT  OF  THE  SUSPENDING   RODS   BEING  TAKEN  INTO   AC- 

COUNT. 


Conceive  the  suspending  rods  to  be  replaced  by  a  con- 


514  THE    SUSPENSION   BRIDGE 

tinuous  flexible  lamina  or  plate  connecting  the  roadway  with 
the  chain,  and  of  such  a  uniform  thickness  that  the  material 
contained  in  it  may  be  precisely  equal  in  weight  to  the  ma- 
terial of  the  suspending  rods.  It  is  evident  that  the  condi- 
tions of  the  equilibrium  will,  on  this  hypothesis,  be  very 
nearly  the  same  as  in  the  actual  case.  Let  f/<8  represent  the 

weight  of  each  square  foot  of  this  plate,  then  will  j*8  /  yd% 

represent  the  weight  of  that  portion  of  it  which  is  suspended 
from  the  portion  DP  of  the  chain,  and  the  whole  load  u  upon 
that  portion  of  the  chain  will  be  represented  by 


....  (622). 
It  may  be  shown,  as  before  (Art.  400.),  that 

(623). 


fKds=—  f(c*+u*)d%.      Substituting    in  equation  (622), 

differentiating  in  respect  to  a?,  and  observing  that  •=-=-  —  , 

ax    c  a/y 


du    udu    m 

tereTy^ 

Transposing,  reducing,  and  assuming, 


=« 


-j 

ay 

A  linear  equation  in  u*,  the  integration  of  which  by  a  well 
known  method  gives 

—2  ay 


Assuming  the  length  of  the  shortest  connecting  rod  DC  to 
be  represented  by  &,  integrating  between  the  limits  b  and  y, 
and  observing  that  when  y=&,  ^=0, 


*  Church's  Int.  Cal.     Art.  176. 


OF   GREATEST    STRENGTH.  515 

-2a&        -2ayv 


-y)  +       +*H-ls  (•          -1)    •  (626). 


Substituting  this  value    of   u*    in  equation  (623),   and 
reducing, 


by  which  expression  the  variation  of  the  section  of  the  chain 
of  uniform  strengtli  is  determined. 

Differentiating  the   equation  -^=-  in  respect  to  a?,  and 

ci  ?/ 
substituting  for  -r-  its  value  from  equation  (624:). 


Substituting  for  wa  its  value  from  equation  (626), 


//7/ 

Multiplying  both  sides  of  this  equation  by  jag,  and  integrat- 
ing between  the  limits  5  and  y,  observing  that  when  y=5, 


Now  let  it  be  observed,  that  the  value  of  T,  being  in  all 
practical  cases  exceedingly  great  as  compared  with  the 
values  of  f*,  and  m,  the  value  of  a  (equation  625)  is  exceed- 
ingly small  ;  so  that  we  may,  without  sensible  error,  assume 
those  terms  of  the  series  s2a(y~&)  which  involve  powers  of 
2a(y—  b)  above  the  first,  to  vanish  as  compared  with  unity, 


*  Church's  Int.  Cal.     Art.  140. 


516  THE   SUSPENSION   BRIDGE. 

This  supposition  being  made,  we  have  s2a(y-&)_ l=2a(y— 5), 
whence,  by  substitution  and  reduction, 


\WMf 

Extracting  the  square  root  of  both  sides,  transposing,  and 
integrating. 


integrating 


the  equation  to  a  parabola  whose  vertex  is  in  D,  and  its 
axis  vertical.* 

The  values  a  and  H  of  a?  and  y  at  the  points  of  suspension 
being  substituted  in  this  equation,  and  it  being  solved  in 
respect  to  0,  we  obtain 


by  which  expression  the  tension  c  upon  the  lowest  point  of 
the  curve  is  determined,  and  thence  the  length  y  of  the  sus- 
pending rod  at  any  given  distance  x  from  the  centre  of  the 
span,  by  equation  (628),  and  the  section  K  of  the  chain  at 
that  point  by  equation  (627),  which  last  equation  gives  by  a 
reduction  similar  to  the  above 


(630). 


405.  The  section  of  the  chains  being  of  uniform  dimensions^ 
as  in  the  common  suspension  bridge,  it  is  required  to 
determine  the  conditions  of  the  equilibrium.^ 

The  weight  of  the  suspending  rods  being  neglected,  and 
the  same  notation  being  adopted  as  in  the  preceding  arti- 
cles, except  that  JA,  is  taken  to  represent  the  weight  of  one 
foot  in  the  length  of  the  chains  instead  of  a  bar  one  square 
inch  in  section,  we  have  by  equation  (614),  since  K  is  here 
constant, 

(631). 


*  Church's  Analyt.  Geom.  Art.  191. 

f  This  problem  appears  first  to  have  been  investigated  by  Mr.  Hodgkinson 
in  the  fifth  volume  of  the  Manchester  Transactions  ;  his  investigation  extends 
to  the  case  in  which  the  influence  of  the  weights  of  the  suspending  rods  is 
included. 


THE   COMMON    SUSPENSION   BRIDGE. 


517 


Differentiating  this  equation  in  respect  to  a?,  and  observing 
that  ~=(l  +  §£)*=(l  +  7)*  (equation  615),  and  that 


du  _du  cty  __du  u  _ 

==~~-^-    " 


/cdu  r         udM 

0  fs(c>+<0*+fV>'y  =-/  f^+tO'+lV' 

Tlie  former  of  these  equations  may  be  rationalised  by 
assuming  (tf+tf^—G  +  zu,   and   the  latter  by  assuming 
u*y=z  ;  there  will  thus  be  obtained  by  reduction 


The  latter  equation  may  be  placed  under  the  form 


which  expression  being  integrated  and  its  value  substituted 
for  z,  we  obtain 


y=- 


The  method  of  rational  fractions  (Church's y.  

Art.  135)  being  applied  to  the  function  under  the  integral 
sign  in  the  former  equation,  it  becomes 


• 


The  integral  in  the  first  term  in  this  expression  is  repre- 
sented by  i  log.s  (j-—  J,  and  that  of  the  second  term  by 


518  RUPTURE   BY   COMPRESSION. 


f*,  ing. 

-*1*  8' 


according  as  ^  is  greater  or  less  than  M-2,  or  according  as  the 
weight  of  each  foot  in  the  length  of  the  chains  is  greater  or 
less  than  the  weight  of  each  foot  in-  the  length  of  the  road- 
way. 

Substituting  for  z  its  value,  we  obtain,  therefore,  in  the 
two  cases, 


_  e   (  .         fo-c)-H^+oa)*  _          ^ 

~Ji  \    g>e  KM-K+'')*     W-fi 


(M-A*0*H-fr«-A«i)*  {(u'+c^-c]  ) 

'£  (M^i^-^-^i^iV-H1)*-*}"  ) 


(638). 


°g 


If  the  given  values,  ^  and  H,  of  a?  and  y  at  the  points  of 
suspension,  be  substituted  in  equations  (633)  and  (632), 
equations  will  be  obtained,  whence  the  value  of  the  constant 
c  and  of  u  at  the  points  of  suspension  may  be  determined  by 
approximation.  A  series  of  values  of  w,  diminishing  from 
the  value  thus  found  to  zero,  being  substituted  in  equations 
(633)  and  (632),  as  many  corresponding  values  of  x  and  y 
will  then  become  known.  The  curve  of  the  chains  may  thus 
be  laid  down  with  any  required  degree  of  accuracy. 

This  common  method  of  construction,  which  assigns  a 
uniform  section  to  the  chains,  is  evidently  false  in  principle ; 
the  strength  of  a  bridge,  the  section  of  whose  chains*  varied 
according  to  the  law  established  in  Art.  401.  (equation  619), 
would  be  far  greater,  the  same  quantity  of  iron  being 
employed  in  its  construction. 


KTTPTURE  BY  COMPRESSION. 

406.  It  results  from  the  experiments  of  Mr.  Eaton  Hodg- 
kinson,*  on  the  compression  of  short  columns  of  different 
heights  but  of  equal  sections,  first,  that  after  a  certain  height 
is  passed  the  crushing  pressure  remains  the  same,  as  the 

*  Seventh  Report  of  the  British  Association  of  Science. 


KUPTUKE   BY   COMPRESSION.  519 

heights  are  increased,  until  another  height  is  attained,  when 
they  begin  to  break  ;  not  as  they  have  done  before,  by  the 
sliding  of  one  portion  upon  a  subjacent  portion,  but  by 
bending.  Secondly,  that  the  plane  of  rupture  is  always 
inclined  at  the  same  constant  angle  to  the  base  of  the 
column,  when  its  height  is  between  these  limits.  These  two 
facts  explain  one  another  ;  for  if  K  represent  the  transverse 
section  of  the  column  in  square  inches,  and  a  the  constant 
inclination  of  the  plane  of  rupture  to  the  base,  then  will 
K  sec.  a  represent  the  area  of  the  plane  of  rupture.  So  that 
if  7  represent  the  resistance  opposed,  by  the  coherence  of 
the  material,  to  the  sliding  of  one  square  inch  upon  the  sur- 
face of  another,*  then  will  7K  sec.  a  represent  the  resistance 
which  is  overcome  in  the  rupture  of  the  column,  so  long  as 
its  height  lies  between  the  supposed  limits  ;  which  resist- 
ance being  constant,  the  pressure  applied  upon  the  summit 
of  the  column  to  overcome  it  must  evidently  be  constant. 
Let  this  pressure  be  represented  by  P,  and  let  CD 
be  the  plane  of  rupture.  Now  it  is  evident  that 
the  inclination  of  the  direction  of  P  to  the  perpen- 
dicular QK  to  the  surface  of  the  plane,  or  its 
equal,  the  inclination  a  of  CD  to  the  base  of  the 
column,  must  be  greater  than  the  limiting  angle 
of  resistance  of  the  surfaces  ;  if  it  were  not,  then 
would  no  pressure  applied  in  the  direction  of  P 
be  sufficient  to  cause  the  one  surface  to  slide  upon  the  other, 
even  if  a  separation  of  the  surfaces  were  produced  along 
that  plane. 

Let  P  be  resolved  into  two  other  pressures,  whose  direc- 
tions are  perpendicular  and  parallel  to  the  plane  of  rupture  ; 
the  former  will  be  represented  by  P  cos.  a,  and  the  friction 
resulting  from  it  by  P  cos.  a  tan.  9  ;  and  the  latter,  repre- 
sented by  P  sin.  a,  will,  when  rupture  is  about  to  take  place, 
be  precisely  equal  to  the  coherence  K/  sec.  a  of  the  plane  of 
rupture  increased  by  its  friction  P  cos.  a  tan.  9,  or  P  sin. 
a=K/  sec.  a  +  P  cos.  a  tan.  9,  whence  by  reduction 


p_      K?  cos.  9        _         2K/  cos.  9 

sin.  (a  —  9)  cos.  a     sin.  (2a  —  9)  —  sin.  9 

It  is  evident  from  this  expression  that  if  the  coherence  of 
the  material  were  the  same  in  all  directions,  or  if  the  unit  of 

*  The  force  necessary  to  overcome  a  resistance,  such  as  that  here  spoken 
of,  has  been  appropriately  called  by  Mr.  Hodgkinson  the  force  necessary  to 
shear  it  across. 


THE  PLANE  OF  RUPTURE. 


coherence  7  opposed  to  the  sliding  of  one  portion  of  the 
mass  upon  another  were  accurately  the  same  in  every  direc- 
tion in  which  the  plane  CD  may  be  imagined  to  intersect 
the  mass,  then  would  the  plane  of  actual  rupture  be  inclined 
to  the  base  at  an  angle  represented  by  the  formula 


since  the  value  of  P  would  in  this  case  be  (equation  634) 
a  minimum  when  sin.  (2a  —  $)  is  a  maximum,  or  when 

If  If       0) 

2a—  9=-,  or  0,=--}--  ;  whence  it  follows  that  a  plane  in- 

£  4:       2i 

clined  to  the  base  at  that  angle  is  that  plane  along  which  the 
rupture  will  first  take  place,  as  P  is  gradually  increased  be- 
yond the  limits  of  resistance. 

The  actual  inclination  of  the  plane  of  rupture  was  found 
in  the  experiments  of  Mr.  Hodgkinson  to  vary  with  the  ma- 
terial of  the  column.  In  cast  iron,  for  instance,  it  varied 
according  to  the  quality  of  the  iron  from  48°  to  58°*,  and 
was  different  in  different  species.  By  this  dependence  of 
the  angle  of  rupture  upon  the  nature  of  the  material,  it  is 
proved  that  the  value  of  the  modulus  of  sliding  coherence 
7  is  not  the  same  for  every  direction  of  the  plane  of  rup- 
ture, or  that  the  value  of  9  varies  greatly  in  different  quali- 
ties of  cast  iron. 

Solving  equation  (634)  in  respect  to  7  we  obtain 

P 
7=07-  sin.  (a—  9)  cos.  a  sec.  9  .....  (636)  ; 

from  which  expression  the  value  of  the  modulus  7  may  be 
determined  in  respect  to  any  material  whose  limiting  angle 
of  resistance  9  is  known,  the  force  P  producing  rupture, 
under  the  circumstances  supposed,  being  observed,  and  also 
the  angle  of  rupture.  f 


THE    SECTION   OF   RUPTURE   IN   A   BEAM. 

407.  When  a  beam  is  deflected  under  a  transverse  strain, 

*  Seventh  Report  of  British  Association,  p.  349. 

f  A  detailed  statement  of  the  results  obtained  in  the  experiments  of  Mr. 
Hodgkmson  on  this  subject  is  contained  in  the  Appendix  to  the  "  Illustrations 
of  Mechanics  "  by  the  author  of  this  work. 


GENERAL  CONDITIONS  OF  RUPTURE. 


521 


the  material  on  that  side  of  it  on  which  it  sustains  the  strain 
is  compressed,  and  the  material  on  the  opposite  side 
extended.  That  imaginary  surface  which  separates  the 
compressed  from  the  extended  portion  of  the  material  is 
called  its  neutral  surface  (Art.  354.),  and  its  position  has 
been  determined  under  all  the  ordinary  circumstances  of 
flexure.  That  which  constitutes  the  strength  of  a  beam  is 
the  resistance  of  its  material  to  compression  on  the  one  side 
of  its  neutral  surface,  and  to  extension  on  the  other  ;  so  that 
if  either  of  these  yield  the  beam  will  be  broken. 

The  section  of  rupture  is  that  transverse  section  of  the 
beam  about  which,  in  its  state  bordering  upon  rupture,  it  is 
the  most  extended,  if  it  be  about  to  yield  by  the  extension 
of  its  material,  or  the  most  compressed  if  about  to  yield  by 
the  compression  of  its  material. 

In  a  prismatic  beam,  or  a  beam  of  uniform  dimensions,  it 
is  evidently  that  section  which  passes  through  the  point  of 
greatest  curvature  of  the  neutral  line,  or  the  point  in 
respect  to  which  the  radius  of  curvature  of  the  neutral  line 
is  the  least,  or  its  reciprocal  the  greatest. 


GENERAL  CONDITIONS  OF  THE  RUPTUKE  OF  A  BEAM. 

408.  Let  PQ  be  the  section  of  rupture  in  a  beam  sustain- 
ing any  given  pressures,  whose 
resultants  are  represented,  if 
they  be  more  in  number  than 
three,  by  the  three  pressures  P1? 
P2,  P3.  Let  the  beam  be  upon 
the  point  of  breaking  by  the 
yielding  of  its  material  to  exten- 
sion at  the  point  of  greatest  ex- 
tension P  ;  and  let  R  represent, 
in  the  state  of  the  beam  border- 
ing upon  rupture,  the  intersection  of  the  neutral  surface 
with  the  section  of  rupture ;  which  intersection  being  in 
the  case  of  rectangular  beams  a  straight  line,  and  being  in 
fact  the  neutral  axis,  in  that  particular  position  which  is 
assumed  by  it  when  the  beam  is  brought  into  its  state  bor- 
dering upon  rupture,  may  be  called  the  axis  of  rupture  ; 
AK  the  area  in  square  inches  of  any  element  of  the  section 
of  rupture,  whose  perpendicular  distance  from  the  axis  of 
rupture  R  is  .represented  by  p ;  S  the  resistance  in  pounds 


522  GENERAL   CONDITIONS    OF   RUPTURE 

opposed  to  the  rupture  of  each  square  inch  of  the  section  at 
P ;  ^  and  ca  the  distances  PR  and  QR  in  inches. 

The  forces  opposed  per  square  inch  to  the  extension  and 
compression  of  the  material  at  different  points  of  the  sec- 
tion of  rupture  are  to  one  another  as  their  several  perpen- 
dicular distances  from  the  axis  of  rupture,  if  the  elasticity 
of  the  material  be  supposed  to  remain  perfect  throughout 
the  section  of  rupture,  up  to  the  period  of  rupture. 

Now  at  the  distance  cl  the  force  thus  opposed  to  the 
extension  of  the  material  is  represented  per  square  inch  by 
S  ;  at  the  distance  p  the  elastic  force  opposed  to  the  exten- 
sion or  compression  of  the  material  (according  as  that 
distance  is  measured  on  the  extended  or  compressed  side),  is 

R 

therefore  represented  per  square  inch  by  — p,  and  the  elastic 

ci 

force  thus  developed  upon  the  element  AK  of  the  section  of 

S 

rupture  by  — pAK,  so  that  the  moment  of  this  elastic  force 
Gi 

o 

about  R  is  represented  by  —  paAK,  and  the  sum  of  the  mo- 

ci 

ments  of  all  the  elastic  forces  upon  the  section  of  rupture 

Q 

about  the  axis  of  rupture  by  — 2p2AK  ;*  or  representing  the 

moment  of  inertia  of  the  section  of  rupture  about  the  axis 
of  rupture  by  I,  the  sum  of  the  moments  of  the  elastic 
forces  upon  the  section  of  rupture  about  its  axis  of  rupture 

Q-r 

is  represented,  at  the  instant  of  rupture,  by  — ,f     Now  the 

ci 

elastic  forces  developed  upon  PQ  are  in  equilibrium  with 
the  pressures  applied  to  either  of  the  portions  APQD  or 
BPQC,  into  which  the  beam  is  divided  by  that  section ;  the 
sum  of  their  moments  about  the  point  P  is  therefore  equal 
to  the  moment  of  R>  about  that  point.  Representing, 
therefore,  by  pl  the  perpendicular  let  fall  from  the  point  B 
upon  the  direction  of  Pn  we  have 

*  It  will  be  observed,  as  in  Art.  358.,  that  the  elastic  forces  of  extension 
and  those  of  compression  tend  to  turn  the  surface  of  rupture  in  the  same 
direction  about  the  axis  of  rupture. 

f  This  expression  is  called  by  the  French  writers  the  moment  of  rupture  ; 
the  beam  is  of  greater  or  less  strength  under  given  circumstances  according 
as  it  has  a  greater  or  less  value. 


BY   TKANSVEKSE   STEAIN.  523 


409.  If  the  deflexion  "be  small  in  the  state  bordering  upon 
rupture,  and  the  directions  of  all  the  deflecting  pressures  be 
perpendicular  to  the  surface  of  the  beam,  the  axis  of  rupture 
passes  through  the  centre  of  gravity  of  the  section,  and  the 
value  of  ol  is  known.  Where  these  conditions  do  not  obtain, 
the  value  of  cl  might  be  determined  by  the  principles  laid 
down  in  Arts.  355.  and  381.  This  determination  would, 
however,  leave  the  theory  of  the  rupture  of  beams  still  in- 
complete in  one  important  particular.  The  elasticity  of  the 
material  has  been  supposed  to  remain  perfect,  at  every  point 
of  the  section  of  rupture,  up  to  the  instant  when  rupture  is 
about  to  take  place.  Now  it  is  to  be  observed,  that  by  rea- 
son of  its  greater  extension  about  the  point  P  than  at  any 
other  point  of  the  section  of  rupture,  the  elastic  limits  are 
there  passed  before  rupture  takes  place,  and  before  they  are 
attained  at  points  nearer  to  the  axis  of  rupture ;  the  forces 
opposed  to  the  extension  of  the  material  cannot  therefore  be 
assumed  to  vary,  at  all  points  of  PR,  accurately  as  their  dis- 
tances from  the  point  R,  in  that  state  of  the  equilibrium  of 
the  beam  which  immediately  precedes  its  rupture  ;  and  the 
sum  of  their  moments  cannot  therefore  be  assumed  to  be  ac- 

QT 

curately  represented  by  the  expression  — .  This  remark  af- 
fects, moreover,  the  determination  of  the  values  of  h  and  R 
(Arts.  355.  and  381.),  and  therefore  the  value  of  cl 

To  determine  the  influence  upon  the  conditions  of  rupture 
by  transverse  strain  of  that  unknown  direction  of  the  insistent 
pressures,  and  that  variation  from  the  law  of  perfect  elasti- 
city which  belongs  to  the  state  bordering  upon  rupture,  we 
must  fall  back  upon  experiment.  From  this  it  has  resulted, 
in  respect  to  rectangular  beams,  that  the  error  produced  by 
these  different  causes  in  equation  (637)  will  be  corrected  if 
a  value  be  assigned  to  ct  bearing,  for  each  given  material,  a 
constant  ratio  to  the  distance  of  the  point  P  from  the  centre 
of  gravity  of  the  section  of  rupture  ;  so  that  c  representing 
the  depth  of  a  rectangular  beam,  the  error  will  be  corrected, 
in  respect  to  a  beam  of  any  material,  by  assigning  to  cl  the 
value  rajc,  where  m  is  a  certain  constant  dependent  upon 
the  nature  of  the  material.  It  is  evident  that  this  cor- 
rection is  equivalent  to  assuming  c,=ic,  and  assigning 
to  S  the  value  ^S  instead  of  that  which  it  has  hitherto 


524 


GENERAL   CONDITIONS    OF   KUPTTJKE1 


been  supposed  to  represent,  viz.  the  tenacity  per  square  inch 
of  the  material  of  the  beam. 

It  is  customary  to  make  this  assumption.  The  values  of  S 
corresponding  to  it  have  been  determined,  by  experiment, 
in  respect  to  the  materials  chiefly  used  in  construction,  and 
will  be  found  in  a  table  at  the  end  of  this  work.  It  is  to 
these  tables  that  the  values  represented  by  S  in  all  subse- 
quent formulae  are  to  be  referred. 


410.  From  the  remarks  contained  in  the  preceding  article, 
it  is  not  difficult  to  conceive  the  existence  of  some  direct  re- 
lation between  the  conditions  of  rupture  by  transverse  and  by 
longitudinal  strain.  Such  a  relation  of  the  simplest  kind  ap- 
pears recently  to  have  been  discovered  by  the  experiments 
of  Mr.  E.  Hodgkinson*,  extending  to  the  conditions  of  rup- 
ture by  compression,  and  common  to  all  the  different  varie- 
ties of  material  included  under  each  of  the  following  great 
divisions — timber,  cast  iron,  stone,  glass. 

The  following  tables  contain  the  summary  given  by  Mr. 
Hodgkinson  of  his  results : — 


Description  of  Material. 

Assumed  Crushing 
Strength  per  Square 
Inch. 

Mean  Tensile 
Strength  per  Square 
Inch. 

Mean  Transverse 
Strength  of  a  Bar 
1  Inch  Square  and 
1  Foot  Long. 

Timber  .... 
Cast-iron         ... 
Stone,  including  marble  - 
Glass  (plate  and  crown)  - 

1000 
1000 
1000 
1000 

1900 
158 
100 
123 

85-1 
19-8 
9-8 

10- 

The  following  table  shows  the  uniformity  of  this  ratio  in 
respect  to  different  varieties  of  the  same  material : — 


Description  of  Material. 

Assumed  Crushing 
Strength  per  Square 
Inch. 

Mean  Tensile 
Strength  per  Square 
Inch. 

Mean  Transverse 
Strength  of  a  Bar 
1  Inch  Square  arid 
1  Foot  Long. 

Black  marble  ... 
Italian  marble 
Rochdale  flagstone  - 
High  Moorstone     - 
Yorkshire  flag 
Stone  from  Little  Hulton, 

near  Bolton 

«  — 

1000 
1000 
1000 
1000 
1000 

I        1000 

143 
84 
104 
100 

70 

10-1 

10-6 
9-9 

9-5 
S, 

*  This  discovery  was  communicated  to  the  British  Association  of  Science  at 
their  meeting  in  1842  ;  it  opens  to  us  a  new  field  of  theoretical  research. 


THE   STRONGEST  FORM  OF   SECTION.  525 


411.    THE    STRONGEST    FORM    OF    SECTION    AT   ANY   GIVEN    POINT 
IN   THE   LENGTH   OF   THE   BEAM. 

Since  the  extension  and  the  compression  of  the  material 
are  the  greatest  at  those  points  which  are  most  distant  from 
the  neutral  axes  of  the  section,  it  is  evident  that  the  mate- 
rial cannot  be  in  the  state  bordering  upon  rupture  at  every 
point  of  the  section  at  the  same  instant  (Art.  388.),  unless  all 
the  material  of  the  compressed  side  be  collected  at  the  same 
distance  from  the  neutral  axis,  and  likewise  all  the  material 
of  the  extended  side,  or  unless  the  material  of  the  extended 
side  and  the  material  of  the  compressed  side  be  respectively 
collected  into  two  geometrical  lines  parallel  to  the  neutral 
axis :  a  distribution  manifestly  impossible,  since  it  would 
produce  an  entire  separation  of  the  two  sides  of  the  beam. 

The  nearest*  practicable  approach  to  this  form  of  section  is 
that  represented  in  the  accompanying  figure,  where  the 
material  is  shown  collected  in  two  thin  but  wide  flanges, 
united  by  a  narrow  rib. 

13  That  which  constitutes  the  strength  of  the 
beam  being  the  resistance  of  its  material  to  com- 
pression on  the  one  side  of  its  neutral  axis,  and 
its  resistance  to  extension  on  the  other  side,  it  is 
evidently  (Art.  388.)  a  second  condition  of  the 
3  strongest  form  of  any  given  section  that  when 
the  beam  is  about  to  break  across  that  section  by 
extension  on  the  one  side,  it  may  be  about  to  break  by  com- 
pression on  the  other.  So  long,  therefore,  as  the  distribution 
of  the  material  is  not  such  as  that  the  compressed  and 
extended  sides  would  yield  together,  the  strongest  form  of 
section  is  not  attained.  Hence  it  is  apparent  that  the 
strongest  form  of  the  section  collects  the  greater  quantity 
of  the  material  on  the  compressed  or  the  extended  side  of 
the  beam,  according  as  the  resistance  of  the  material  to 
compression  or  to  extension  is  the  less.  Where  the  material 
of  the  beam  is  cast  iron*,  whose  resistance  to  extension  is 
greatly  less  than  its  resistance  to  compression,  it  is  evident 
that  the  greater  portion  of  the  material  must  be  collected  on 
the  extended  side. 

Thus,  then,  it  follows,  from  the  preceding  condition  and 

*  It  is  only  in  cast  iron  beams  that  it  is  customary  to  seek  an  economy  of 
the  material  in  the  strength  of  the  section  of  the  beam  ;  the  same  principle  of 
economy  is  surely,  however,  applicable  to  beams  of  wood. 


526 


THE   STRONGEST   FORM   OF    SECTION. 


this,  that  the  strongest  form  of  section  in  a  cast  iron  beam  is 
that  by  which  the  material  is  collected  into  two  unequal 
flanges  joined  by  a  rib,  the  greater  flange  being  on  the 
extended  side ;  and  the  proportion  of  this  inequality  of  the 
flanges  being  just  such  as  to  make  up  for  the  inequality  of 
the  resistances  of  the  material  to  rupture  by  extension  and 
compression  respectively. 

Mr.  Hodgkinson,  to  whom  this  suggestion  is  due,  has 
directed  a  series  of  experiments  to  the  determination  of  that 
proportion  of  the  flanges  by  which  the  strongest  form  of 
section  is  obtained.* 

The  details  of  these  experiments  are  found  in  the  following 
table:- 


Number  of 
Experiment. 

Ratio  of  the  Sections 
of  the  Flanges. 

Area  of  whole 
Section  in  Square 
Inches. 

Strength  per  Square 
Inch  of  Section  in  Ibs. 

1 

1  to  1- 

2-82 

2368 

2 

1  to  2- 

2-87 

2567 

3 

1  to  4- 

3-02 

2737 

4 

1  to  4-5 

3-37 

3183 

5 

1  to  5-5 

5-0 

3346 

6 

1  to  6-1 

6-4 

4075 

In  the  first  five  experiments  each  beam  broke  by  the  tear- 
ing asunder  of  the  lower  flange.  The  distribution  by  which 
both  were  about  to  yield  together — that  is,  the  strongest 
distribution — was  not  therefore  up  to  that  period  reached. 
At  length,  however,  in  the  last  experiment,  the  beam  yielded 
by  the  compression  of  the  upper  flange.  In  this  experiment, 
therefore,  the  upper  flange  was  the  weakest ;  in  the  one  be- 
fore it,  the  lower  flange  was  the  weakest.  For  a  form 
between  the  two,  therefore,  the  flanges  were  of  equal  strength 
to  resist  extension-  and  compression  respectively  ;  and  this 
was  the  strongest  form  of  section  (Art.  388.). 

In  this  strongest  form  the  lower  flange  had  six  times  the 
material  of  the  upper.  It  is  represented  in  the  accompany- 
ing figure. 

A  In  the  best  form  of  cast  iron  beam  or 

girder  used  before  these  experiments, 
there  was  never  attained  a  strength  of 
more  than  2885  Ibs.  per  square  inch  of 
section.  There  was,  therefore,  by  this 
form,  a  gain  of  1190  Ibs.  per  square  inch 
of  the  section,  or  of  fths  the  strength  of 
the  beam. 


*  Memoirs  of  Manchester  Philosophical  Society,  vol.  iv.  p.  453. 
tions  of  Mechanics,  Art.  68. 


Illustro- 


THE  BEAM  OF  GREATEST  STRENGTH.          527 


412.  THE  SECTION  OF  RUPTURE. 

The  conditions  of  rupture  being  determined  in  respect  to 
any  section  of  the  beam  by  equation  (637),  it  is  evident  that 
the  particular  section  across  which  rupture  will  actually  take 
place  is  that  in  respect  to  which  equation  (637}  is  first  satis- 
fied, as  P,  is  continually  increased  ;  or  that  section  in  respect 
to  which  the  formula 

— (638) 

ftft 

is  the  least. 

If  the  beam  be  loaded  along  its  whole  length,  arid  x  repre- 
sent the  distance  of  any  section  from  the  extremity  at  which 
the  load  commences,  and  f*  the  load  on  each  foot  of  the 
length,  then  (Art.  371.)  Pj?,  is  represented  by  4*aja.  The 
section  of  rupture  in  this  case  is  therefore  that  section  in 
respect  to  which  M-  is  first  made  to  satisfy  the  equation. 

QT 

—  ;  or  in  respect  to  which  the  formula 


is  the  least. 

If  the  section  of  the  beam  be  uniform,  —  is  constant ;  the 

G\ 

section  of  rupture  is  therefore  evidently  that  which  is  most 
distant  from  the  free  extremity  of  the  beam. 

4:13.  THE  BEAM  OF  GREATEST  STRENGTH. 

The  beam  of  greatest  strength  being  that  (Art.  388.)  which 
presents  an  equal  liability  to  rupture  across  every  section,  or 
in  respect  to  which  every  section  is  brought  into  the  state 
bordering  upon  rupture  by  the  same  deflecting  pressure,  is 
evidently  that  by  which  a  given  value  of  Pis  made  to  satisfy 
equation  (637)  for  all  the  possible  values  of  I,  pl9  and  cl9  or 
in  respect  to  which  the  formula 

^T (64°) 

is  constant. 


THE    STRENGTH   OF   BEAMS. 


If  the  beam  be  uniformly  loaded  throughout  (Art.  371.), 
this  condition  becomes 


or  constant,  for  all  points  in  the  length  of  the  beam. 


414.   ONE  EXTREMITY  OF  A  BEAM   is  FIRMLY  IMBEDDED  IN 

MASONRY,  AND  A  PRESSURE  IS  APPLIED  TO  THE  OTHER 
EXTREMITY  IN  A  DIRECTION  PERPENDICULAR  TO  ITS  LENGTH! 
TO  DETERMINE  THE  CONDITIONS  OF  THE  RUPTURE. 

If  x  represent  the  distance  of  any  section  of  the  beam 
from  the  extremity  A  to  which  the  load  P 
is  applied,  and  a  its  whole  length,  and  if  the 
section  of  the  beam  be  everywhere  the 
same,  then  the  formula  ( 638 )  is  least 
at  the  point  B,  where  x  is  greatest:  at 
this  point,  therefore,  the  rupture  of  the 
beam  will  take  place.  Representing  by 
P  the  pressure  necessary  to  break  the 
beam,  and  observing  that  in  this  case  the 
perpendicular  upon  the  direction  of  P 
from  the  section  of  rupture  is  represented 
by  #,  we  have  (equation  637) 

P=-|^ (642). 

If   the  section  of   the   beam  be  a  rec- 
tangle, whose  breadth  is  &  and  its  depth  0, 

j  1  T  -      ""       ~  — 

then  1= 


(643). 


If  the  beam  be  a  solid  cylinder,  whose  radius  is  0,  then 
(Art.  364.)  I=frc\  c.—c. 

G*         .  .  (644). 


a 


If  the  beam  be  a  hollow  cylinder,  whose  radii  are  rl  and 
r,,  I=^f(r*—r*)  ;  which  expression  may  be  put  under  the 
form  fliw^-f  ic*)  (see  Art.  86.),  r  representing  the  mean 


THE  STRENGTH  OF  BEAMS.  529 

radius  of  the  hollow  cylinder,  and  c  its  thickness.     Also 


.-.P=«rS 


(645). 


415.  The  strongest  form  of  beam  under  the  conditions  sup- 
posed in  the  last  article. 


1st.  Let  the  section  of  the  beam  be  a 
rectangle,  and  let  y  be  the  depth  of 
this  rectangle  at  a  point  whose  distance- 
from  its  extremity  A  is  represented  by 
#,  and  let  its  breadth  5  be  the  same 
throughout.  In  this  case  I— rV^V 
cl=^y:  therefore  (equation  637)  P=. 
SI  jf 
—  =±Sb— .  If  therefore,  P  be  taken 

CjC  X 

to  represent  the  pressure  which  the 
beam  is  destined  just  to  support,  then 
the  form  of  its*  section  ABC  is  deter- 
mined (Art.  413.)  by  the  equation 

6P 


it  is  therefore  a  parabola,  whose  vertex 
is  at  A.* 

If  the  portion  DO  of  the  beam:  do  not  rest  against 
masonry  at  every  point,  but  only  at  its 
extremity  D,  its  form;  should;  evidently  be 
the  same  with  that  of  ABC: 

2d.  Let  the  section,  be  a  circle,  and 
let  y  represent  its  radius  at  distance  x 
from  its  extremity  A,  then  I= 

y' 


c^=y\     therefore    P=J*S~  so   that   the 

oc 

geometrical  form  of  its  longitudinal 
section  is  determined  by  the  equa- 
tion 


*  The  portion  of  the  beam  imbedded  in  the  masonry  should  have  the  form 
described  in  Art.  417. 

34 


530 


THE    STRENGTH    OF   BEAMS. 


(647), 


P  representing  the  greatest  pressure  to  which  it  is  destined 
to  be  subjected- 


416.  THE  CONDITIONS  OF  THE  RUPTURE  OF  A  BEAM  SUPPORTED 
AT  ONE  EXTREMITY,  AND  LOADED  THROUGHOUT   ITS   WHOLE 

LENGTH. 

Kepresenting  the  weight  resting  upon  each  inch  of  its 

length  a  by  M.,  and  observ- 
ing that  the  moment  of  the 
weight  upon  a  length  <  of 
the  beam  from  A,  about  the 
corresponding  neutral  axis, 
is  represented  (Art.  371.) 
by  %we\  it  is  apparent  (Art. 
412.)  that,  if  the  beam  be 
of  uniform  dimensions,  its 
section  of  rupture  is  BD. 
Its  strength  is  determined 
by  substituting  ^a?  for  P  j^ 
in  equation  (637),  and  solving  in  respect  to  ^ ;  we  thus  obtain 


2SI 


(648); 


by  which  equation  is  determined  the  uniform  load  to  which 
the  beam  may  be  subjected,  on  each  inch  of  its  length. 

For  a  rectangular  beam,  whose  width  is  5  and  its  depth 
c,  this  expression  becomes 


417.  To  determine  the  form  of  greatest  strength  (Art.  413.) 
in  the  case  of  a  beam  having  a  rectangular  section  of  uni- 
form breadth  •^a?2  must  be  substituted  for  PJp1  in  equation 


(637),  and 
reduction 


for  I,  and  \y  for  cl  ;    whence  we  obtain  by 


(650. 


THE   STRENGTH   OF   BEAMS. 


531 


The  form  of  greatest  strength  is  therefore,  in  this  case,  the 
straight  line  joining  the  points  A  and  B  ;  the  distance  DB 
being  determined  by  substituting  the  distance  AD  for  x  in 
the  above  equation. 

That  portion  BED  of  the  beam  which  is  embedded  in  the 
masonry  should  evidently  be  of  the  same  form  with  DBA.* 


418.  If,  in  addition  to  the  uniform  load  upon  the  beam,  a 
given  weight  "W"  be  suspended  from  A,  Jf*aj*-|-"Waj  must  be 


substituted  for  P  j>,  in  equation  (637)  ;  we  shall  thus  obtain 
lor  the  equation  to  the  form  of  greatest  strength 


which  is  the  equation  to  an  hyperbola  having  its  vertex 
at  A.f 

*  It  is  obvious  that  in  all  cases  the  strength  of  a  beam  at  each  point  of  its 
length  is  dependent  upon  the  dimensions  of  its  cross  section  at  that  point,  and 
that  its  general  form  may  in  any  way  be  changed  without  impairing  its  strength 
provided  those  dimensions  of  the  section  be  everywhere  preserved. 

f  Church's  Anal.  Geom.     Art.  124. 


532 


THE   STRENGTH  OF  BEAMS. 


419.  THE  BEAM  OF  GREATEST  STRENGTH  IN  REFERENCE  TO  THE 

FORM     OF     ITS     SECTION     AND     TO     THE     VARIATION     OF  THE 

DIMENSIONS     OF     ITS      SECTION,     WHEN     SUPPORTED     AT  ONE 

EXTREMITY    IN    A    HORIZONTAL    POSITION,    AND     LOADED  UNI- 
FORMLY  THROUGHOUT   ITS   LENGTH, 


The  general  form  of  the  section  must  evidently  be  that 

described  in  Art.  411.  Let 
the  same  notation  be  taken 
as  in  Art.  365.,  ^xcept  that 
the  depth  MQ  of  the  plate 
or^  rib  joining  the  two 
flanges  is  to  be  represented 
by  2/,  and  its  thickness  by  c. 


therefore  by  equation  (503), 


so  that  dz=y,  and  A3=cy  ; 


Also  representing  by  cl  the  distance  of  the  centre  of  gravity 
of  the  whole  section  from  the  upper  surface  of  the  beam, 
we  have  cl(AJ  +  Az  +  cy)=(jiy+d,)cy+(y  +  d,+%d1)Al+%dt 
Aa.  Substituting  for  I  and  cl  in  equation  (637),  and  for  ¥lpl  its 
value  -J-fAa?2,  x  being  taken  to  represent  the  distance  AM,  and 
M.  the  load  on  each  inch  of  tliat  length,  we  have  (Art. 
413.) 

3f*  f 

In®  = 


) 
d, 


(y  +  Zdjoy  +  2(y  +  d,  4-  K)  A,  +  A  & 

.....  (652). 

Let  the  area  cy  of  the  section  of  the  rib  now  be  neglected, 
as  exceedingly  small  when  compared  with  the  areas  of  the 
sections  of  the  flanges,  an  hypothesis  which  assigns  to  the 
beam  somewhat  less  than  its  actual  strength  ;  let  also  the 
area  of  the  section  of  the  upper  flange  be  assumed  equal  to 
n  times  that  of  the  lower,  or  Aa=^A1, 


(653). 


If  the  flanges  be  exceedingly  thin,  dl  and  d^  are  exceed- 
ingly small  and  may  be  neglected.     The  equation  will  then 


THE  STRENGTH  OF  BEAMS.  533 

become  that  to  a  parabola  whose  vertex  is  at  A  and  its  axis 
vertical.  This  may  therefore  be  assumed  as  a  near  approxi- 
mation to  the  true  form  of  the  curve  AQC. 

Where  the  material  is  cast  iron,  it  appears  by  Mr.  Hodg- 
kinson's  experiments  (Art.  411.)  that  n  is  to  be  taken=6. 


420.  A  BEAM  OF  UNIFORM  SECTION  IS  SUPPORTED  AT  ITS 
EXTREMITIES  AND  LOADED  AT  ANY  POINT  BETWEEN  THEM! 
IT  IS  REQUIRED  TO  DETERMINE  THE  CONDITIONS  OF  RUPTURE. 

The  point  of  rupture  in  the  case   of  a  uniform   section 

is  evidently  (Art.  412.)  the  point 
C,  from  which  the  load  is  sus- 
pended; representing  AB,  AC, 
EC,  by  a,  a^  and  #a;  and  ob- 
serving that  the  pressure  Pt 

Wa 

upon  the  point  B  of  the  beam  = -,  so  that  the  moment 

of  Pj,  in  respect  to  the  section  of  rupture  C  = — *,  we 


SI 

have,  by  equation  (637),  "  ~1^a=_ 

a         GI 


(654). 


If  the  beam  be  rectangular ,  I =-^50*,  <?!=•£•<?, 
•VW==5  T-£ (655); 

O    tJjQ/2 

where  "W  represents  the  breaking  weight,  S  the  modulus  of 
rupture,  a  the  length,  b  the  breadth,  c  the  depth,  and  al9  a% 
the  distances  of  the  point  C  from  the  two  extremities,  all 
these  dimensions  being  in  inches. 

If  the  load  be  suspended  in  the  middle,  a1=a,t=^ay 

"W=¥^  .....(656). 


If  the  beam  be  a  solid  cylinder,  whose  radius  =<?,  then  1= 
Jtf<?4,  <?!=<?;  therefore,  equation  (654), 

W=^  -^  .  .  (657). 


534  THE  STRENGTH  OF  BEAMS. 

• 

If  the  beam  be  a  hollow  cylinder,  whose  mean  radius  is  r, 
and  its  thickness  c,  I=tcr(ri+%c*),  c1^=r-\-^c]  therefore, 
equation  (654), 

( 


If  the  section  of  the  beam  be  that  represented  in  Art.  411., 
being  everywhere  of  the  same  dimensions,  then,  observing 
that  Acl=^d3Aa-}-dsA^  nearly,  we  have,  (equations  503  and 
654) 


l12 
-T~  (2A1+A,)a1a,4 

where  A15  A3  represent  the  areas  of  the  sections  of  the  upper 
and  lower  flanges,  and  A3  that  of  the  connecting  rib  or  plate, 
and  dv  d»  da  their  respective  depths. 


421.  A  BEAM  IS  SUPPORTED  AT  ITS  EXTREMITIES,  AND  LOADED 
AT  ANT  GIVEN  POINT  BETWEEN  THEM  J  ITS  SECTION  IS  OF  A 
GIVEN  GEOMETRICAL  FORM,  BUT  OF  VARIABLE  DIMENSIONS  : 
IT  IS  REQUIRED  TO  DETERMINE  THE  LAW  OF  THIS  VARIA- 
TION, SO  THAT  THE  STRENGTH  OF  THE  BEAM  MAY  BE  A 
MAXIMUM. 

W  representing  the  breaking  load  upon  the  beam,  and 

#„  «2  the  distances  of  its  point 
of  suspension  C,  from  A  and 
B,  the  pressure  P,  upon  A  is 


represented  by  2.  If,  there- 

a 

fore   (Art.    388.),  x  represent 

the  horizontal  distance  of  any  section  MQ  from  the  point  of 
support  A,  and  I  its  moment  of  inertia,  and  ^  the  distance 
from  its  centre  of  gravity  to  the  point  where  rupture  is  about 
to  take  place  (in  this  case  its  lowest  point) ;  then  by  equa- 
tion (637) 

W"<~-SI  .  (660). 


a 


1st.  Let  the  section  be  rectangular  ;  let  its  breadth  b  be 
constant;  and  let  its   depth  at  the  distance  x  from  A  be 


THE   STRENGTH   OF   BEAMS.  535 

represented  by  y  ;  therefore  I=i-V^y'>  ci—%y-     Substituting 
in  the  above  equation  and  reducing, 


The  curve  AC  is  therefore  a  parabola,  whose  vertex  is  at 
A,  and  its  axis  horizontal.  In  like  manner  the  curve  EC  is 
a  parabola,  whose  equation  is  identical  with  the  above,  ex- 
cept that  al  is  to  be  substituted  in  it  for  ay 

2d.  Let  the  section  of  the  beam  be  a  circle.  Represent- 
ing the  radius  of  a  section  at  distance  x  from  A  by  y,  we 
have  I^Jtf^4,  c1=y.)  therefore  by  equation  (660) 

(662> 

3d.  Let  the  section  of  the  beam  be  circular  ;  but  let  it  be 
hollow,  the  thickness  of  its  material  being  every  where  the 
same,  and  represented  by  o.  If  y=  mean  radius  of  cylinder 
at  distance  x  from  A,  then  I=<7r<3?/(y2-f  Jc2),  c1= 


422.    THE    BEAM   OF    GREATEST    ABSOLUTE    STRENGTH    WHEN 
LOADED  AT  A  GIVEN  POINT  AND  SUPPORTED  AT  THE  EXTRE- 

MITIES. 


Let  the  section  of  the  beam  be  that  of  greatest  strength 
rt.  411.).     Substituting  in  equation  (66 
as  before  in  equation  (652),  and  reducing, 


(Art.  411.).     Substituting  in  equation  (660)  the  value  of  — 


&.  .  (664). 

If  the  section  cy  of  the  rib  be  every  where  exceedingly 
small  as  compared  with  the  sections  of  the  flanges,  and  if 

j^B=OH-9w+^o+i*/ ( 


There  is  a  value  of  x  in  this  equation  for  which  y  becomes 


536  THE   STRENGTH   OF   BEAMS. 

impossible.  For  values  less  than  this,  the  condition  of  uni- 
form strength  cannot  therefore  obtain.  It  is  only  in  respect 
to  those  parts  of  the  beam  which  lie  between  the  values  of 
SB  (measured  from  the  two  points  of  support)  for  which  y 
thus  becomes  impossible,  that  the  condition  of  greatest 
strength  (Art.  388.)  is  possibly.  If  its  proper  value  be 
assigned  to  n  (Art.  411.),  this  may  be  assumed  as  an  approxi- 
mation to  the  true  form  of  beam  of  THE  GREATEST  ABSOLUTE 
STRENGTH.  When  the  material  is  cast  iron,  it  appears  by  the 
experiments  of  Mr.  Hodgkinson  (Art.  411.)  that  n=6.  A2 
represents  in  all  the  above  cases  the  section  of  the  extended 
flange  ;  in  this  case,  therefore,  it  represents  the  section  of 
the  lower  flange. 

The  depth  CD  at  the  point  of  suspension  may  be  deter- 
mined by  substituting  a^  for  x  in  equation  (665)  ;  its  value  is 
thus  found  to  be  represented  by  the  formula 


(666), 


423.  If  instead  of  the  depth  of  the  beam  being  made  to 
vary  so  as  to  adapt  itself  to  the  condition  (Art.  388.)  of  uni- 
form strength,  its  breadth  b  be  made  thus  to  vary,  the  depth 
c  remaining  the  same  ;  then,  assuming  the  breadth  of  the 
upper  flange  at  the  distance  x  from  the  point  of  support  A 
to  be  represented  by  y,  and  the  section  of  the  lower  flange 
to  be  n  times  greater  than  that  of  the  upper;  observing, 
moreover,  that  in  equation  (503)  At=  yd^  Az=nA.1=nyd1  ; 
neglecting  also  A3  as  exceedingly  small  when  compared  with 
A,  and  Aa,  and  writing  c  for  <#3,  we  have  by  reduction, 


ftrr.l 

Also  <?,  being  the  distance  of  the  lower  surface  of  the  beam 
from  the  common  centre  of  gravity  of  the  sections  of  the 
two  fl^iges,  we  have  cl(n-\-\)=c.  Eliminating,  therefore, 
the  values  of  I  and  c,  from  equation  (660), 

«=         \  A(»+l)  (d^nd.^+ncd,     y  .....  (667), 


the  equation  to  a  straight  line.  Each  flange  is  therefore  in 
this  case  a  quadrilateral  figure,  whose  dimensions  are  deter- 
mined from  the  greatest  breadth  ;  this  last  being  known,  foi 


THE   STRENGTH   OF   BEAMS. 


537 


the  upper  flange,  by  substituting  ^  for  x  in  the  above  equa- 
tion, and  solving  in  respect  of  y,  and  for  the  lower  flange 
from  the  equation  nb^—b^  in  which  &j,  £9  represent  the 
greatest  breadths  of  the  two  flanges,  and  aa  d^  their  depths 


424.  A  BEAM  IS  LOADED  UNIFORMLY  THROUGHOUT  ITS  WHOLfl 
LENGTH,  AND  SUPPORTED  AT  ITS  EXTREMITIES  I  IT  IS  REQUIRED 
TO  DETERMINE,  1.  TfiE  CONDITIONS  OF  ITS  RUPTURE  WHEN  ITS 
CROSS  SECTION  IS  UNIFORM  THROUGHOUT  ;  2.  THE  STRONGEST 

FORM  OF  BEAM  HAVING  EVERY  WHERE  A  RECTANGULAR  CROSS 
SECTION  ;  3.  THE  BEAM  OF  GREATEST  STRENGTH  IN  REFER- 
ENCE BOTH  TO  THE  FORM  AND  THE  VARIATION  OF  ITS  CROSS 
SECTION. 


1.  If  the  section  of  the  beam  be  uniform,  its  point  of  rup- 
ture is  determined  by  formula  (639) 
to  be  its  middle  point.  Kepresenting, 
therefore,  in  this  case,  the  length  of 
the  beam  by  2«,  the  weight  on  each 
inch  of  its  length  by  ^  and  its  breadth 
by  & ;  and  observing  that  in  this  case 
p^— /xa2—  ^a?=%t*<a\  we  have  by 
equation  (637) 


n: 


281 


(668), 


where  f*  represents  the  load  per  inch  of  the  length  of  the 
beam  necessary  to  produce  rupture.  In  the  case  of  a  rectan- 
gular beam,  this  equation  becomes 

P  =  %£.        ..(669). 


2.  To  determine  the  form  of  the  beam  of  greatest  strength 
having  a  rectangular  section  of 
given  breadth  &,  let  y  be  taken  to 
represent  its  depth  PQ  at  a  point  P, 
and  x  its  horizontal  distance  from 
the  point  A.  Then  I  =  T\fy/\ 
c1=^/'j  also  Pj?x  (equation  637) 

representing  the  moment  of  the  resultant  of  the  pressures 
upon  AP  about  the  centre  of  gravity  of  ~PQ=^ax 
therefore  by  equation  (637)  pax— - 


538 


THE   STRENGTH    OF  BEAMS. 


the  equation  to  an  ellipse,  whose  vertex  is  in  A,  and  its 
centre  at  C. 

3.  To  determine  the  beam  of  absolute  maximum  strength, 
let  it  be  assumed,  as  in  Art.  422,  that  the  area  of  the  section 
of  the  rib  is  exceedingly  small  as  compared  with  the  areas 
of  the  sections  of  the  flanges ;  and  let  the  area  of  the  section 
of  the  lower  or  extended  flange  be  n  times  that  of  the  upper ; 

A  4     AW    I     -A-i  J  (7l  +  l)  (d?+nd*)  +  12ny*  ) 
then,  as  in  Art.  422,  -=-JT  )  -  —     \_  ,      , — 

also  Pj?1=M-oa?— if^B8 ;  whence,  by  equation  (637), 


SA 


'  +  2K 


4.  If  it  be  proposed  to  make  the  rib  or  plate  uniting  the 
two  flanges  everywhere  of  the  same  depth,*  and  so  to  vary 
the  breadths  of  the  flanges  as  to  give  to  the  beam  a  uniform 
strength  at  all  points  under  these  circumstances  ;  represent- 
ing by  y  the  breadth  of  the  upper  flange  at  a  horizontal 
distance  x  from  the  point  of  support,  we  shall  obtain,  as  in 
Art.  423, 


Moreover,  ~Plp1=^ax—  J^*==-Jfw?(2a—  a?);  whence  we  obtain 
by  substitution  in  equation  (637),  and  reduction, 


}y  .....  (673)  ; 


the  equation  to  a  parobola,f  whose  axis  is  in  the  horizontaJ 
line  bisecting  the  flange  at  right  angles,  its  parameter  repre- 


*  As  in  Mr.  Hodgkinson's  construction. 
•j-  Church's  Anal.  Geom.     Art.  171. 


THE  STRENGTH  OF  BEAMS.  539 

sented  by  the  coefficient  of  y  in  the  preceding  equation,  and 
half  the  breadth  of  the  flange  in  the  middle  determined  by 
the  formula 


(674) 


The  equation  to  the  lower  flange  is  determined  by  substi- 
tuting for  y,  in  equation  (673),  ^  ;  whence  it  follows  that 
the  breadth  of  the  lower  flange  in  the  middle  is  equal  to 
that  of  the  upper  multiplied  by  the  fraction  -~  . 


425.  A  RECTANGULAR  BEAM  OF  UNIFORM  SECTION,  AND  UNI- 
FORMLY LOADED  THROUGHOUT  ITS  LENGTH,  IS  SUPPORTED  BY 
TWO  PROPS  PLACED  AT  EQUAL  DISTANCES  FROM  ITS  EXTREM- 
ITIES :  TO  DETERMINE  THE  CONDITIONS  OF  RUPTURE. 

It  is  evident  from  formula  (639)  that  the  section  of  rup- 
ture of  the  portion  CA  of  the 
beam  is  at  A,  and  therefore  that 
the  conditions  of  its  rupture  are 
determined  (Art.  416.)  by  the 
equation 

•TO-        -(675); 


where  ^,  represents,  as  before,  the 
load  upon  each  inch  of  the  length  of  the  beam,  b  its 
breadth,  c  its  depth,  and  a,  the  length  of  the  portion  AC. 

Again,  it  is  evident  that  the  point  of  rupture  of  the  por- 
tion AB  of  the  beam  is  at  E.  Now  the  value  of  P^, 
(equation  637)  is,  in  respect  to  the  portion  AE  of  the  beam, 
\^^a(a — $j) — i"^^2  5  ^a  representing  the  whole  length  of  the 
beam  ^  the  load  upon  each  inch  of  the  length  of  the  beam 
which  would  produce  rupture  at  E,  and  therefore  ^a  the 
resistance  of  each  prop  in  the  state  bordering  upon  rupture  } 

also  -=:|foa.      Whence,   by   equation    (637), 

0, 


54:0  THE   STRENGTH   OF  BEAMS. 

4 

426.  THE  BEST  POSITIONS  OF  THE  PROPS. 

If  the  load  i^  be  imagined  to  be  continually  increased,  it 
is  evident  that  rupture  will  eventually  take  place  at  A  or  at 
E  according  as  the  limit  represented  by  equation  (675),  or 
that  represented  by  equation  (676),  is  first  attained,  or 
according  as  f*t  or  f*2  is  the  less. 

Let  M-!  be  conceived  to  be  tue  less,  and  let  the  prop  A  be 
moved  nearer  to  the  extremity  C ;  #t  being  thus  diminished, 
ju-j  will  be  increased,  and  ^  diminished,  Now  if,  after  this 
change  in  the  position  of  the  prop,  f\  still  remains  less  than 
fxa,  it  is  evident  that  the  beam  will  bear  a  greater  load  than 
it  would  before,  and  that  when  by  continually  increasing 
the  load  it  is  brought  into  the  state  bordering  upon  rupture 
at  A  it  will  not  be  in  the  state,  bordering  upon  rupture  at  E. 
The  beam  may  therefore  be  strengthened  yet  further  by 
moving  the  prop  A  towards  C;  and  thus  continually,  so 
that  the  beam  evidently  becomes  the  strongest  when  the 
prop  is  moved  into  such  a  position  that  i^1  may  just  equal 
f*2.  This  position  is  readily  determined  from  equations  (675) 
and  (676)  to  be  that  in  which 

~ (677). 


427.  A  RECTANGULAR  BEAM  OF  UNIFORM  SECTION  AND  UNI- 
FORMLY LOADED  IS  SUPPORTED  AT  ITS  EXTREMITIES,  AND  BY 
TWO  PROPS  SITUATED  AT  EQUAL  DISTANCES  FROM  THEM  :  TO 
DETERMINE  THE  CONDITIONS  OF  RUPTURE. 

Adopting  the  same  notation  as  in  Art.  374.,  it  appears  by 

equation  (543)  that  the  dis- 
tance x,  of  the  point  of  great- 
est curvature  of  the  neutral 
line,  and  therefore  of  the  sec- 
tion of  rupture  in  AB  from 


==    A    (Art,    407.)    being    that 
where  -T-,  -t*  is  the  greatest,  is 

^  CUtHj 

determined  by  the   equation 

*  The  curvature  of  the  neutral  line  being  everywhere  exceedingly  small, 
^  may  be  assumed  =1.     The  expression  for  the  radius  of  curvature  in  terms 

of  the  rectangular  co-ordinates  resolves  itself  therefore,  in  this  case,  into  the 
second  differential  coefficient. 


THE    STRENGTH   OF   BEAMS. 


541 


f/tf^P,  it  being  observed  that,  at  the  section  of  rupture,  the 
neutral  line  is  concave  to  the  axis  of  #,  and  therefore  the 
second  differential  coefficient  (equation  543)  negative.  The 
value  of  P  is  that  determined  by  equation  (551)  ;  so  that 


n(8»-8) 


(678)' 


where  a  represents  the  distance  AE,  and  na  the  distance  AB. 
Let  P  represent  the  intersection  of  the  neutral  line  with 
the  plane  of  rupture,  and  i^  the  load  per  inch  of  the  whole 
length  of  the  beam  which  would  produce  a  rupture  at  P. 
Now  the  sum  of  the  moments  of  the  forces  impressed  on 
AP  (other  than  the  elastic  forces  on  the  section  of  rupture) 
is  represented  in  the  state  bordering  upon  rupture,  by 

P^— i^a?,8;  or,  since  P1=i*1aj1,  it  is  represented  by^— Pa2; 

J/xi 

whence  it  follows  by  equation  (637)  that  the  conditions  of 
the  rupture  of  the  beam  between  A  and  B  are  determined 

by  the  eq nation  •?-—  P.2 =jSfo2,  or, 
2f*t 

....  (679). 


Eliminating  the  value  of 
(679),  we  obtain 


between  equations  (551)  and 


8n(2»-3) 


r 


(680). 


Substituting    this  value    of   f*,   in   equation   (679),   and 
reducing 

'-3)          } (681). 


If  the  points  B  and  C 
coincide,  or  the  beam  be 
supported  by  a  single  prop 
in  the  middle,  n=l;  there- 
fore, by  equations  (680)  and 
(681), 


(682); 


54:2  THE    STRENGTH    OF   BEAMS. 


Similarly,  it  appears  by  equation  (547)  that  the  point  of 
greatest  curvature  between  B  and  C  is  E  ;  if  the  rupture  of 
the  beam  take  place  first  between  these  points,  it  will  there- 
fore take  place  in  the  middle.  Let  ^  represent  the  load, 
per  inch  of  the  length,  which  would  produce  a  rupture  at  E. 
Now,  the  sum  of  the  moments  about  E  of  the  forces  im- 
pressed upon  AE  is  Pl($-hP2(a—  no)—  foi>9a*=(P  ' 
rjia—  iiv&9=  M-X—  (^a—  P,)^a—  if^X  (since  P1  + 
Therefore  by  equation  (637) 

+  I>i^=iS5ca  .....  (684). 

Substituting  for  ~Pl  its  value  from  equation  (551),  and 
solving  in  respect  to  ^2, 

S6c'          271-3  , 

( 


If  the  load  be  continually  increased,  the  beam  will  break 
between  A  and  B,  or  between  B  and  C,  according  as  f*, 
(equation  680)  or  f*a  (equation  685)  is  the  less. 

428.  THE  BEST  POSITIONS  OF  THE  PROPS. 

It  may  be  shown,  as  in  Art.  426.,  that  the  positions  in 
which  the  props  must  be  placed  so  as  to  cause  the  beam  to 
bear  the  greatest  possible  load  distributed  uniformly  over  its 
whole  length,  are  those  by  which  the  values  of  f*,  (equation 
680)  and  ^  (equation  685)  are  made  equal  ;  the  former  of 
these  quantities  representing  the  load  per  inch  of  the  length, 
which  being  uniformly  distributed  over  the  whole  beam 
would  just  produce  rupture  between  A  and  B,  if  it  did  not 
before  take  place  between  B  and  C  ;  and  the  latter  that 
which  would,  under  the  same  circumstances,  produce  rup- 
ture between  B  and  C  if  it  had  not  before  taken  place 
between  A  and  B. 

Let,  then,  na  represent  the  distance  at  which  the  prop  B 
must  be  placed  from  A  to  produce  this  equality  ;  and  let  the 
value  of  M-,  given  by  equation  (679)  be  substituted  for  (xa  in 
equation  (684)  ;  we  shall  thus  obtain  by  reduction 


*  3(1-27*)-  9(1-27*) 
Solving  this  quadratic  in  respect  to  P,a, 


THE    STRENGTH    OF   BEAMS.  54:3 


The  negative  sign  must  be  taken  in  this  expression,  since 
the  positive  would  give  P1=f*1a  by  equation  (679),  and  cor- 
responds therefore  to  the  case  n=Q.  Assuming  the  negative 
sign,  and  reducing,  we  have  3(2n—  l)Pla=Sbc*.  Substitut- 
ing in  this  expression  for  Px  its  value  from  equation  (681), 
and  reducing, 

—  1)  (2n—  3) 
~ 


The  three  roots  of  this  equation  are  1-57087,  '61078,  and 
•26994.  The  first  and  last  are  inadmissible  ;  the  one  carry- 
ing the  point  B  beyond  E,  and  the  other  assigning  to  Pj  a 
negative  value.*  the  best  position  of  the  prop  is  therefore 
that  which  is  determined  by  the  value 

n=  -61078  .....  (686). 


429.    THE  CONDITIONS   OF  THE  RUPTURE   OF  A   RECTANGULAR 

BEAM  LOADED  UNIFORMLY  THROUGHOUT  ITS  LENGTH,  AND 
HAVING  ITS  EXTREMITIES  PROLONGED  AND  FIRMLY  IMBEDDED 
IN  MASONRY. 

It  has  been  shown  (Art.  376.)  that  the  conditions  of  the 
deflexion  of  the  beam  are,  in  this  case,  the  same  as  though 
its  extremities,  having  been  prolonged  to  a  point  A  (see  Jig. 
p.  540.),  such  that  AB  might  equal  '6202  AE,  had  been  sup- 
ported by  a  prop  at  B,  and  by  the  resistance  of  any  fixed 
surface  at  A.  The  load  which  would  produce  the  rupture 
of  the  beam  is  therefore,  in  this  case,  the  same  as  that  which 
would  produce  the  rupture  of  a  beam  supported  by  props 
(Art.  427.)  between  the  props,  and  is  determined  by  that 
value  of  f*a  (equation  685)  which  is  given  by  the  value  '6202 
of  n.  It  is,  however,  to  be  observed  that  the  symbol  a 
§ 

*  We  may,  nevertheless,  suppose  the  extremity  A,  instead  of  being  sup 
ported  from  beneath,  to  be  pinned  down  by  a  resistance  or  a  pressure  acting 
from  above.  This  case  may  occur  in  practice,  and  the  best  position  of  the 
props  corresponding  to  it  is  that  whic^h  is  determined  by  the  least  root  of  the 
equation,  viz.  '26994. 


544 


THE   STRENGTH   OF   COLUMNS. 


represents  in  that  equation 
the  distance  AE  (Jig.  Art. 
427.)  ;  and  that  if  we  take 
it  to  represent  the  distance 
BE  in  that  or  the  accompa- 
nying figure,  we  must  sub- 

stitute =  --  for  a  in  equa- 
1—n 

tion  (685),  since  0=BE=AE—  AB=(1—  w)AE  ;  «so  that 
AE=:j—  -.  This  substitution  being  made,  equation  (685), 
becomes 


|Cj 

h- 

—  [— 

i  — 

—  — 

i  —  p_ 

1 

^J    1    '     .- 

~l 

i 

—  ,— 

C1 

K 

U 

1 

1         1         1 

1        1         1 

i 

1        1         1 

1 

i 

\ 

1        1         1 

II         i 

i 

\ 

J        1        1 

_      II         i 

\ 

\ 

_l—JL__L__j 

and  substituting  the  value  *6202  for  n,  we  obtain  by  reduc- 
tion 


(687), 


by  which  formula  the  load  per  inch  of  the  length  of  the 
beam  necessary  to  produce  rupture  is  determined. 

If  the  beam  had  not  been  prolonged  beyond  the  points  of 
support  B  and  C  and  imbedded  in  the  masonry,  then  the 
load  per  inch  of  the  length  necessary  to  produce  rupture 
would  have  been  represented  by  equation  (669)  :  eliminat- 
ing between  that  equation  and  equation  (687),  we  obtain 
|xa=3fx  ;  so  that  the  load  per  inch  of  the  length  necessary  to 
produce  rupture  is  3  times  as  great,  when  the  extremities 
of  the  beam  are  prolonged  and  firmly  imbedded  in  the  ma- 
sonry, as  when  they  are  free  ;  i.  e.  the  strength  of  the  beam 
is  3  times  as  great  in  the  one  case  as  in  the  other. 


430.  THE  STRENGH  OF  COLUMNS. 

For  all  the  knowledge  of  this  subject  on  which  any  reli- 
ance .can  be  placed  by  the  engineer  he  is  endebted  to  expe- 
riment.* 

*  The  hypothesis  upon  which  it  has  been  customary  to  found  tlfe  theoretical 
discussion  of  it,  is  so  obviously  insufficient,  and  the  results  have  been  shown 
by  Mr.  Hodgkinson  to  be  so  little  in  accordance  with  those  of  practice,  that 
the  high  sanction  it  has  received  from  labours  such  as  those  of  Euler,  Lagrange, 
Poisson,  and  Navier,  can  no  longer  establish  for  it  a  claim  to  be  admitted 
among  the  conclusions  of  science.  (See*  Appendix  K.) 


THE    STRENGTH   OF   COLUMNS. 


545 


The  following  are  the  principal  results  obtained  in  the 
valuable  series  of  experimental  inquiries  recently  instituted 
by  Mr.  Eaton  Hodgkinson.* 


FORMULAE  REPRESENTING  THE  ABSOLUTE  STRENGTH  OF  A  CYL- 
INDRICAL COLUMN  TO  SUSTAIN  A. PRESSURE  IN  THE  DIRECTION 
OF  ITS  LENGTH. 

D= external  diameter  or  side  of  the  square  of  the  column 
in  inches. 

D1=internal  diameter  of  hollow  cylinder  in  inches.. 

L— length  in  feet. 

W= breaking  weight  in  tons. 


Nature  of  the  Column. 

Both  Ends  being 
rounded,  the  Length  of 
the  Column  exceeding 
fifteen  times  its 
Diameter. 

Both  Ends  being  flat,,  the 
Length  of,  the  Column 
exceeding-  thirty  times  i.ts 
Diameter.. 

Solid    cylindrical    column  of  ) 
cast  iron    -        -        -        -  ) 
Hollow  cylindrical  column  of  ) 

W=U-9^ 
D""-^"" 

7)3.68, 

^=44-16^ 

T)3-SB_D    3.66 

W      44-^4. 

cast  iron     -        -        -        -  ) 
Solid    cylindrical    column   of  ) 

L1'7 

JJ3.78 

w  —  42'fl 

4       L" 

JJS.66 

"W  —  ll^'*?^ 

wrought  iron      -        -           ) 

Solid  square  pillar  of  Dantzic  ) 
oak  (dry)    -         -        -        -  f 
Solid  square  pillar  of  red  deal  ) 
(dry)  J 

La 

W=10'95?j 

W=n7'81^ 
JL 

In  all  cases  the  strength  of  a  column,  one  of  whose  ends 
was  rounded  and  the  other  flat,  was  found  to  be  an  arith- 
metic mean  between  the  strengths  of  two  other  columns  of 
the  same  dimensions,  one  having  both  ends  rounded  and  the 
other  having  both  ends  flat. 

The  above  results  only  apply  to  the  case  in  which  the 
length  of  the  column  is  so  great  that  its  fracture  is  produced 
wholly  by  the  bending  of  its  material ;  this  limit  is  fixed  by 
Mr.  'Hodgkinson  in  respect  to  columns  of  cast  iron  at  about 
fifteen  times  the  diameter  when  the  extremities  are  rounded, 

*  From  a  paper  by  Mr.  Hodgkinson,  published  in  the  second  part  of  the 
Transactions  of  the  Royal  Society  for  1840,  to  which  the  royal  medal  of  the 
Society  was  awarded.  The  experiments  were  made  at  the  expense  of 
Mr.  Fairbairn  of  Manchester,  by  whose  liberal  encouragement  the  researches 
of  practical  science  have  been  in  other  respects  so  greatly  advanced. 

35 


546  TORSION. 

and  thirty  times  ths  diameter  when  they  arc  flat.  In 
shorter  columns  fracture  takes  place  partly  by  the  crushing 
and  partly  by  the  bending  of  the  material.  To  these  shorter 
columns  the  following  rule  was  found  to  apply  with  suf- 
ficient accuracy  :  —  "  If  W,  represent  the  weight  in  tons 
which  would  break  the  column  by  bending  alone  (or  if  it 
did  not  crush)  as  given  by  the  preceding  formula,  and  W2 
the  weight  in  tons  which  would  break  the  column  by  crush- 
ing alone  (or  if  it  did  not  bend)  as  determined  from  the 
preceding  table,  then  the  actual  breaking  weight  "W  of  the 
column  is  represented  in  tons  by  the  formula 


Columns  enlarged  in  the  middle.  It  was  found  that  the 
strengths  of  columns  of  cast  iron,  whose  diameters  were  from 
one  and  a  half  times  to  twice  as  great  in  the  middle  as  at 
the  extremities,  were  stronger  by  one  seventh  than  solid 
columns,  containing  the  same  quantity  of  iron  and  of  the 
same  length,  when  their  extremities  were  rounded;  and 
stronger  by  one  eighth  or  one  ninth  when  their  extremities 
were  flat  and  rendered  immoveable  by  discs. 


431.  EELATIVE  STRENGTH  OF  LONG  COLUMNS  OF  CAST 

WROUGHT   IRON,    STEEL,  AND   TIMBER   OF   THE    SAME   DIMENSIONS. 

• — Calling  the  strength  of  the  cast  iron  column  1000,  the 
strength  of  the  wrought  iron  column  wi  ll,  according  to  these 
experiments,  be  1745,  that  of  the  cast  steel  column  2518,  of 
the  column  of  Dantzic  oak  108*8,  and  of  the  column  of  red 
deal  78-5. 

Effect  of  drying  on  the  strength  of  columns  of  timber. — 
It  results  from  these  experiments,  that  the  strength  of  short 
columns  of  wet  timber  to  resist  crushing  is  not  one  half  that 
of  columns  of  the  same  dimensions  of  dry  timber. 


TORSION. 

432.  The  elasticity  of  torsion. 
Let  ABCD  represent  a  solid  cylinder,  ou<e  -of  whose  trans- 


TORSION, 


54:7 


verse  sections  AEB  is  immoveably  fixed, 
and  every  other  displaced  in  its  own  plane, 
about  its  centre,  by  the  action  of  a  pres- 
sure P  applied,  at  a  given  distance  a  from 
the  axis,  to  the  section  CD  of  the  cylinder 
in  the  plane  of  that  section  and  round  its 
centre ;  thet  cylinder  is  said,  under  these 
circumstances,  to  be  subjected  to  torsion, 
and  the  forces  opposed  to  the  alteration  of 
its  form,  and  to  its  rupture,  constitute  its 
resistance  to  torsion. 

Let  aabfi  be  any  section  of  the  cylinder 
whose  distance  from  the  section  AEB  is 
represented  by  a?,  and  let  a/3  represent  that 
diameter  of  the  section  aabfi  which  was 
parallel  to  the  diameter  AB  before  the  torsion  commenced : 
let  ab  be  the  projection  of  the  diameter  AB  upon  the  sec- 
tion aalfi,  and  let  the  angle  aca  be  represented  by  6. 

Now  the  elastic  forces  called  into  action  upon  the  section 
aaibfi  are  in  equilibrium  with  the  pressure  P.  But  these 
elastic  forces  result  from  the  displacement  of  the  section 
aab@  upon  its  immediately  subjacent  section.  Moreover, 
the  actual  displacement  of  any  small  element  AK  of  the 
section  aabfi,  upon  the  subjacent  section,  evidently  depends 
partly  upon  the  angular  displacement  of  the  one  section 
upon  the  other,  and  partly  upon  the  distance  p  of  the 
element  in  question  from  the  axis  of  the  cylinder.  Now  the 
angle  oca  or  6  is  evidently  the  sum  of  the  angular  displace- 
ments of  all  the  sections  between  aabfi  and  AEB  upon  their 
subjacent  sections ;  and  the  angular  displacement  of  each 
upon  its  subjacent  section  is  the  same,  the  circumstances 
affecting  the  displacement  of  each  being  obviously  the  same  : 
also  the  number  of  these  sections  varies  as  a?,  and  the  sum 
of  their  angular  displacements  is  represented  by  6 ;  there- 
fore the  angular  displacement  of  each  section  upon  its  sub- 

& 
jacent  section  varies  as  -,  and  the  actual  displacement  of 

& 
the  small  element  AK  of  the  section  a&bfi  varies  as  -p.     Now 

the  material  being  elastic,  the  pressure  which  must  be 
applied  to  this  element  in  order  to  keep  it  in  this  state  of 
displacement  varies  as  the  amount  of  the  displacement 

(Art.  345.),  or  as  -p.     Let  its  actual  amount,  when  referred 


548  TOESION. 

8 
to  a  unit  of  surface,  be  represented  by  G-p,  where  G  is  a 

certain  constant  dependant  for  its  amount  on  the  elastic 
qualities  of  the  material,  and  called  the  modulus  of  torsion ; 
then  will  the  force  of  torsion  required  to  keep  the  element 

& 
AK  iii  its  state  of  displacement  be  represented  by  G-pAK,  and 

$ 
its  moment  about  the  axis  of  the  cylinder  by  G-  paAK.     So 

that  the  sum  of  the  moments  of  all  such  forces  of  torsion  in 
respect  to  the  whole  section  aubfi  will  be  represented  by 

A  A 

G  -  2p2AK,  or  by  G  -I,  if  I  represent  the  moment  of  inertia 

of  the  section  about  the  axis  of  the  cylinder.  Now  these 
forces  are  in  equilibrium  with  P ;  therefore,  by  the  principle 
of  the  equality  of  moments, 

P0=GI- (689). 

<KJ 

If  r  represent  the  radius  of  the  cylinder,  I=farr*  (Art. 
85.).  Substituting  this  value,  representing  by  L  the  whole 
length  of  the  cylinder,  and  by  ©  the  angle  through  which 
its  extreme  section  CD  is  displaced  or  through  which  OP  is 
made  to  revolve,  called  the  angle  of  torsion,  and  solving  in 
r,espect  to  ©, 


Thus,  then,  it  appears  that  when  the  dimensions  of  the 
Y"  cylinder  are  given,  the  angle  of  torsion  ©  varies 
directly  as  the  pressure  P  by  which  the  torsion 
is  produced ;  whence,  also,  it  follows  (Art.  97.) 
that  if  the  cylinder,  after  having  been  deflected 
through  any  distance,  be  set  free,  it  will  oscil- 
late isochronously  about  is  position  of  repose, 
the  time  T  of  each  oscillation  being  represented 
in  seconds  (equation  76)  by  the  formula 


since 


/*T\ 
by  equation  (690)  P=  I  H-FT  I  (®0)  j  m  which  expression 


TORSION.  549 

(©#)  represents  the  length  of  the  path  described  by  the 
point  P  from  its  position  of  repose,  so  that  the  moving  force 
upon  the  point  P,  when  the  pressure  prducing  torsion  is 
removed,  varies  as  the  path  described  by  it  from  its  position 
of  repose. 

The  above  is  manifestly  the  theory  of  Coulomb's  Torsion 
Balance.*  "W  represents  in  the  formula  the  weight  of  the 
mass  supposed  to  be  carried  round  by  the  point  P,  and  the 
inertia  of  the  cylinder  itself  is  neglected  as  exceedingly 
small  when  compared  with  the  inertia  of  this  weight. 

The  torsion  of  rectangular  prisms  has  been  made  the  sub- 
ject of  the  profound  investigations  of  MM.  Cauchyf,  Lame, 
et  Clapeyron£,  and  Poisson.g  It  results  from  these  investi- 
gations 1  that  if  5  and  c  be  taken  to  represent  the  sides  of  the 
rectangular  section  of  the  prism,  and  the  same  notation  be 
adopted  in  other  respects  as  before,  then 


M.  Cauchy  has  shown  the  values  of  the  constant  G  to 
be  related  to  those  of  the  modulus  of  elasticity  E  by  the 
formula 

G=fE (693). 


In  using  the  values  of  G  deduced  by  this  formula  from 
the  table  of  moduli  of  elasticity,  all  the  dimensions  must  be 
taken  in  inches,  and  the  weights  in  pounds. 


433.  ELASTICITY  OF  TORSION  IN  A  SOLED  HAVING  A  CIRCULAR 

SECTION   OF  VARIABLE   DIMENSIONS. 

Let  db  represent  an  element  of  the  solid  contained  by 

*  Illustrations  of  Mechanics,  Art.  37. 

j  Exereices  de  Mathematiques,  4e  annee. 

i  Crelle's  Journal.  §  Memoires  de  1' Academic,  tome  viii. 

f  Navier,  Resume  des  Lecons,  &c.,  Art.  159. 


550 


TOBSION. 


planes,  perpendicular  to  the  axis,  whose  dis- 
tance from  one  another  is  represented  by 
the  exceedingly  small  increment  A&>  of  the 
distance  x  of  the  section  ab  from  the  fixed 
section  AB,  and  let  its  radius  be  repre- 
sented by  y  ;  and  suppose  the  whole  of  the 
solid  except  this  single  element  to  become 
rigid,  a  supposition  by  which  the  conditions 
of  the  equilibrium  of  this  particular  element 
will  remain  unchanged,  the  pressure  P  re- 
maining the  same,  and  being  that  which 
produces  the  torsion  of  this  single  element. 
Whence,  representing  by  A&  the  angle  of 
torsion  of  this  element,  and  considering  it 

a  cylinder  whose  length  is  AOJ,  we  have  by  equation  (689), 

substituting  for  I  its  value 


Passing  to  the  limit,  and  integrating  between  the  limits  0 
and  L,  observing  that  at  the  former  limit  0=0,  and  at  the 
latter  0=0. 


2P0   r 

=-^ - 


.  .  (694.) 


If  the  sides  AC  and  BD  of  the  solid  be  straight  lines,  its 
form  being  that  of  a  truncated  cone,  and  if  rv  and  rz  repre- 
sent its  diameters  AB  and  CD  respectively  ;  then 


Also, 


dx 
dy 


i    ryt 

1       'a 


.. 


TORSION.  551 


4:34.  THE  RUPTURE  OF  A  CYLINDER  BY  TORSION. 

It  is  evident  that  rupture  will  first  take  place  in  respect 
to  those  elements  of  the  cylinder  which  are  nearest  to  its 
surface,  the  displacement  of  each  section  upon  its  subjacent 
section  being  greatest  about  those  points  which  are  nearest 
to  its  circumference.  If,  therefore,  we  represent  by  T  the 
pressure  per  square  inch  which  will  cause  rupture  by  the 
sliding  of  any  section  of  the  mass  upon  its  contiguous  sec- 
tion,* then  will  T  represent  the  resistance  of  torsion  per 
square  inch  of  the  section,  at  the  distance  r  from  the  axis,  at 
the  instant  when  rupture  is  upon  the  point  of  taking  place, 
the  radius  of  the  cylinder  being  represented  by  r.  Whence 
it  follows  that  the  displacement,  and  therefore  the  resistance 
to  torsion  per  square  inch  of  the  section,  at  any  other  dis- 
tance p  from  the  axis,  will  be  represented  at  that  distance  by 

—  ,  the  resistance  upon  any  element  AK,  by  -  p^K,  -and  the 
r  .         r 

sum  of  the  moments  about  the  axis,  of  the  resistances  of  all 

T  T 

such  elements,  by  -  2p2AK,  or  by   -  I,  ®r  substituting  for  I 
r  r 

its  value  (equation  64)  by  -JTV.  But  these  resistances  are 
in  equilibrium  with  the  pressure  P,  which  produces  torsion, 
acting  at  the  distance  a  from  the  axis  ; 

/.Pa=JTW  ____  (696). 

It  results  from  the  researches  of  M.  Cauchy,  before  referred 
to,  that  in  the  case  of  a  rectangular  section  whose  sides  are 
represented  by  ~b  and  <?,  the  conditions  of  rupture  are  deter- 
mined by  the  equation 

(697). 


The  length  of  a  prism  subjected  to  torsion  does  not  -affect 
the  actual  amount  of  the  pressure  required  to  produce  rup- 
ture, but  only  the  angle  of  torsion  (equation  690)  which 
precedes  rupture,  and  therefore  the  space  through  which 

*  Or  the  pressure  per  square  inch  necessary  to  shear  it  across  (Art.  406.). 
\  Navier,  Resume  d'un  Cours,  &c.  Art.  167. 


552  TORSION 


the  pressure  must  be  made  to  act,  and  the  amount  of  WORK 
which  must  be  done  to  produce  rupture. 

According  to  M.  Cauchy,  the  modulus  of  rupture  by  tor- 
sion T  is  connected  with  that  S  of  rupture  by  transverse 
strain  by  the  equation 

T=|S (698). 


r  vi. 

IMPACT* 


435.  THE  IMPACT  OF  TWO  BODIES  WHOSE  CENTRES  OF  GRAVITS 

MOVE   IN   THE    SAME   RIGHT   LINE,   AND   WHOSE   POINT   OF   CON- 
TACT  IS   IN   THAT   LINE. 

From  the  period  when  a  body  first  receives  the  impact  of 
another,  until  that  period  of  the  impact  when  both  move  for 
an  instant  with  the  same  velocity,  it  is  evident  that  the  sur- 
faces must  have  been  in  a  state  of  continually  increasing 
compression :  the  instant  when  they  acquire  a  common  velo- 
city is,  therefore,  that  of  their  greatest  compression.  When 
this  common  velocity  is  attained,  their  mutual  pressures  will 
have  ceased  if  they  be  inelastic  bodies,  and  they  will  move 
with  a  common  motion ;  if  they  be  elastic,  their  surfaces 
will,  in  the  act  of  recovering  their  forms,  be  mutually 
repelled,  and  the  velocities  will,  after  the  impact,  be  dif- 
ferent from  one  another. 


436.  A  BODY  WHOSE  WEIGHT  18  W,,  AND  WHICH  IS  MOVING 
IN  A  HORIZONTAL  DIRECTION  WITH  A  UNIFORM  VELOCITY 
REPRESENTED  BY  "V,,  IS  IMPINGED  UPON  BY  A  SECOND  BODY 
WHOSE  WEIGHT  IS  Wa,  AND  WHICH  IS  MOVING  IN  THE  SAME 
STRAIGHT  LINE  WITH  THE  VELOCITY  Ya  :  IT  IS  REQUIRED  TO 
DETERMINE  THEIR  COMMON  VELOCITY  V  AT  THE  INSTANT  OF 
GREATEST  COMPRESSION. 

Let/,  represent  the  decrement  per  second  of  the  velocity 
of  W,  at  any  instant  of  the  impact  (Art.  94.),  or  rather  the 
decrement  per  second  which  its  velocity  would  experience 
if  the  retarding  pressure  were  to  remain  constant ;  then  will 

*  Note  (u\  Ed,  App. 


554:  IMPACT. 


lfi  represent  (Art.  95.)  the  effective  force  upon  W1  ;  and  if 
j  be  taken  to  represent,  under  the  same  circumstances,  the 
increment  of  velocity  received  by  W,,  then  will  —  ?/a  repre- 

sent the  effective  force  upon  "W,.  Whence  it  follows,  by  the 
principle  of  D'Alembert  (Art.  103.),  that  if  these  effective 
forces  be  conceived  to  be  applied  to  the  bodies  in  directions 
opposite  to  those  in  which  the  corresponding  retardation 
and  acceleration  take  place,  they  will  be  in  equilibrium  with 
the  other  forces  applied  to  the  bodies.  But,  by  supposition, 
no  other  forces  than  these  are  applied  to  the  bodies  :  these 
are  therefore  in  equilibrium  with  one  another, 

w         W 

»W,=^/.  .....  (699). 

y          & 

Let  now  an  exceedingly  small  increment  of  the  time  from 
the  commencement  of  the  impact  be  represented  by  A£,  and 
let  A^  and  A^2  represent  the  decrement  and  increment  of 
the  velocities  of  the  bodies  respectively  during  that  time, 


.-.(Art. 
/.(equation  699)  Wl  .  A-y1=Ws  .  Ava; 

and  this  equality  obtaining  throughout  that  period  of  the 
impact  which  precedes  the  period  of  greatest  compression,  it 
follows  that  when  the  bodies  are  moving  in  the  same  direc- 
tion 

W1(V,-V)=W,(V-VO  .....  (700); 

since  Y,—  -  Y  represents  the  whole  velocity  lost  by  W,  during 
that  period,  and  Y—  Y,  the  whole  velocity  gained  by  "W,. 

If  the  bodies  be  moving  in  opposite  directions,  and  their 
common  motion  at  the  instant  of  greatest  compression  be  in 
the  direction  of  the  motion  of  W,,  then  is  the  velocity  lost 
by  Wx  represented  as  before  by  (Yx—  Y)  ;  but  the  sum  of 
the  decrements  and  increments  of  velocity  communicated  to 
"W2,  in  order  that  its  velocity  Y2  may  in  the  first  place  be 
destroyed,  and  then  the  velocity  Y  communicated  to  it  in  an 
opposite  direction,  is  represented  by  (Ya-f  Y). 


Solving  these  equations  in  respect  to  Y,  we  obtain 


IMPACT.  555 


V— 


the  sign  ±  being  taken  according  as  the  motions  of  tlu 
bodies  before  impact  are  both  in  the  same  direction  or  in 
opposite  directions. 
If  the  second  body  was  at  rest  before  impact,  Ya=0,  and 

wy 

v=^rtt 

If  the  bodies  be  equal  in  weight, 


The  demonstration  of  this  proposition  is  wholly  indepen- 
dent of  any  hypothesis  as  to  the  nature  of  the  impinging 
bodies  or  their  elastic  properties  ;  the  proposition  is  there- 
fore true  of  all  bodies,  whatever  may  be  their  degrees  of 
hardness  or  their  elasticity,  provided  only  that  at  the 
instant  of  greatest  compression  every  part  of  each  body 
partakes  in  the  common  velocities  of  the  bodies,  there  being 
no  relative  or  vibratory  motion  of  the  parts  of  either  body 
among  themselves. 


437.  To  DETERMINE  THE  WORK  EXPENDED  UPON  PRODUCING 
THE  STATE  OF  THE  GREATEST  COMPRESSION  OF  THE  SUR- 
FACES OF  THE  BODIES. 

The  same  notation  being  taken  as  before,  the  whole  work 
accumulated  in   the   bodies,  before  impact,  is  represented 

"W  W2 

by  -J — *  Yxa  4-  •£  — ?Y2a ;  and  the  work  accumulated  in  them 

y  y 

at  the  period  of  greatest  compression,  when  they  move  with 

"W 

the   common   velocity   Y,  is  represented  by  -J- 

"Now  the  difference  between  the  amounts  of  work  accumu- 
lated in  the  bodies  in  these  two  states  of  their  motion  has 
been  expended  in  producing  their  compression ;  if,  there- 
fore, the  amount  of  work  thus  expended  be  represented  bj 
v,  we  have 

WW 

u  V,' +        V.'-f 


556  IMPACT. 


or  substituting  for  V  its  value  from  equation  (701),  and 
reducing, 


U=7T-\ 


This  expression  represents  the  amount  of  work  permanently 
lost  in  the  impact  of  two  inelastic  bodies,  their  common 
velocity  after  impact  being  represented  by  equation  (T01). 
If  W,  be  exceedingly  great  as  compared  with  Wx, 

,)' ....  (704). 


438.    TWO   ELASTIC   BODIES   IMPINGE    UPON  ONE  ANOTHER  :   IT   IS 
REQUIRED   TO   DETERMINE   THE   VELOCITY   AFTER   IMPACT. 

If  the  impinging  bodies  be  perfectly  elastic,  it  is  evident 
that  after  the  period  of  their  greatest  compression  is  passed, 
they  will,  in  the  act  of  expanding  their  surfaces,  exert 
mutual  pressures  upon  one  another,  which  are,  in  corres- 
ponding positions  of  the  surfaces,  precisely  the  same  with 
those  which  they  sustained  whilst  in  the  act  of  compression ; 
whence  it  follows  that  the  decrements  of  velocity  expe- 
rienced by  that  body  whose  motion  is  retarded  by  this 
expansion  of  the  surfaces,  and  the  increments  acquired  by 
that  whose  velocity  is  accelerated,  will  be  equal  to  those 
before  received  in  passing  through  corresponding  positions, 
and  therefore  the  whole  decrements  and  increments  thus 
received  during  the  whole  expansion  equal  to  those  received 
during  the  whole  compression. 

Now  the  velocity  lost  by  W,  during  the  compression  is 
represented  by  (Y,— Y) ;  that  lost  by  it  during  the  expan- 
sion, or  from  the  period  of  greatest  compression  to  that 
when  the  bodies  separate  from  one  another,  is  therefore 
represented  by  the  same  quantity.  But  at  the  instant  of 
greatest  compression  both  bodies  had  the  velocity  Y ;  the 
velocity  vl  of  W,  at  the  instant  of  separation  is  therefore 
Y— (Y!— Y),  or  2Y— Y,.  In  like  manner,  the  velocity 
gained  by  W2  during  compression,  and  therefore  during 
expansion,  being  represented  by  (Y^  Y2),  and  its  velocity 
at  the  instant  of  greatest  compression  by  Y,'its  velocity  vt 
at  the  instant  of  separation  is  represented  by  Y  +  (Y=Fv2), 
or  by  2Y=pYa,  the  sign  =F  being  taken  according  as  the 


IMPACT.  557 

motion  of  the  bodies  before  impact  was  in  the  same    or 
opposite  directions. 

Substituting  for  Y  its  value  in  these  expressions  (equation 
701),  and  reducing,  we  obtain 


i_ii, 
w.+w, 


w.+w, 

If  the  bodies  be  perfectly  elastic  and  ep,ual  in  weight, 
v,=Y,,  ua=Y1  ;  they  therefore,  in  this  case,  interchange  their 
velocities  by  impact  ;  and  if  either  was  at  rest  before  impact, 
the  other  will  be  at  rest  after  impact. 

If  W3  be  exceedingly  great  as  compared  with  "W,,  v^  = 
—  Y1±2Y2,  0a=±Va.  In  this  case  v1  is  negative,  or  the 
motion  of  the  lesser  body  alters  its  direction  after  impact, 
when  their  motions  before  impact  were  in  opposite  direc- 
tions ;  or  when  they  were  in  the  same  direction,  provided 
that  2Y,  be  not  greater  than  Yr 

439.  If  the  elasticities  of  the  balls  be  imperfect,  the  force 
with  which  they  tend  to  separate  at  any  given  point  of  the 
expansion  is  different  from  that  at  the  corresponding  point 
of  the  compression  ;  the  decrements  and  increments  of  the 
velocities,  produced  during  given  corresponding  periods  of 
the  compression  and  expansion,  are  therefore  different  ; 
whence  it  follows  that  the  whole  amounts  of  velocity,  lost 
by  the  one  and  gained  by  the  other  during  the  two  periods, 
are  different  :  let  them  bear  to  one  another  the  ratio  of  1  to  e. 
~Now  the  velocity  lost  during  compression  by  "W,  is  under 
all  circumstances  represented  by  (Ya—  Y);  that  lost  during 
expansion  is  therefore  represented,  in  this  case,  by  e  (Yj—  Y); 
therefore,  vl=Y—  e(Vl—  Y)—  (1  +  0)Y—  eVr  In  like  man- 
ner, the  velocity  gained  by  W2  during  compression  is  in  all 
cases  represented  by  (Y=F  Ya)  ;  that  gained  during  expansion 
is  therefore  represented  by  ^(Y^Y,,);  therefore,  0a=Y-f- 
e(  V  qp  Y,)  =  (1  +  e}  Y  =F  e  Ya.  Substituting  for  Y,  and  reducing, 

.  ,7(m  . 


558  IMPACT. 

440.  IN  THE  IMPACT  OF  TWO  ELASTIC  BODIES,  TO  DETERMINE 
THE  ACCUMULATED  WORK,  OR  ONE  HALF  THE  VIS  VIVA,  LOST 
BY  THE  ONE  AND  GAINED  BY  THE  OTHER. 

The  vis  viva  lost  by  W\  during  the  impact  is  evidently 
represented  by  — LY12 -v*  =  — -fVY— v*)  =  — -  \  Y.a— 

g         9         g  g  \ 

Kl+,)Y_,YirJ^Kl- 


Substituting  in  this  expression  its  value  for  Y  (equation 
701)  reducing  and  representing  by  u^  one  half  the  vis  viva 
lost  by  Wl  in  its  impact,  or  the  amount  by  which  its  accumu- 
lated work  is  diminished  by  the  impact  (Art.  67.), 


1= 


,V,j  ....(709). 


Similarly,  if  u^  be  taken  to  represent  one  half  the  vis  viva 
gained  by  W2,  or  the  amount  by  which  its  accum  ulated  work 
is  increased  by  the  impact,  then 


(710). 


441.  Let  u  be  taken  to  represent  the  whole  amount  of  the 
work  accumulated  in  the  two  bodies  before  their  impact, 
which  is  lost  during  their  impact.  This  amount  of  work  is 
evidently  equal  to  the  difference  between  that  gained  by  the 
one  body  and  lost  by  the  other;  so  that  u=u1—u.l.  Substi- 
tuting the  values  of  u^  and  u^  from  the  preceding  equations, 
and  reducing,  we  obtain 


_ 


This  expression  is  equal  to  one  half  the  vis  viva  lost  during 
the  impact  of  the  bodies.  If  the  bodies  be  perfectly  elastic, 
6=1,  and  i£=0.  In  this  case  there  is  no  real  loss  of  vis  viva 


IMPACT.  559 

in  the  impact,  all  that  which  the  one  body  yields,  during  the 
impact,  being  taken  up  by  the  other.* 

442.  In  the  preceding  propositions  it  has  been  supposed 
that  the  motions  of  the  impinging  body,  and  the  body  im- 
pinged upon,  are  opposed  by  no  resistance  whatever  during 
the  period  of  the  impact.     There  is  no  practical  case  in 
which  this  condition  obtains  accurately.     If,  nevertheless, 
the  resistance  opposed  to  the  motion  of  each  body  be  small, 
as  compared  with  the  pressure  exerted  by  each  upon  the 
other,  at  any  period  of  the  impact,  then  it  is  evident  that  all 
the  circumstances  of  the  impact  as  it  proceeds,  and  the  mo- 
tion of  each  body  at  the  instant  when  it  ceases,  will  be  very 
nearly  the  same  as  though  no  resistance  were  opposed  to  the 
motion  of  either,  f 

443.  As  an  illustration  of  the  principle  established  in  the 
last  article,  let  it  be  required  to  determine  the  space  through 

*  It  has  been  customary,  nevertheless,  to  speak  of  a  loss  of  vis  viva  in  the 
impact  of  perfectly  elastic  bodies.  This  loss  is  in  all  such  cases  to  be  under- 
stood only  as  a  loss  experienced  by  one  of  the  bodies,  and  not  as  an  absolute 
loss.  When  the  impinging  bodies  are  perfectly  elastic,  it  is  evident  that  the 
one  flies  away  with  all  the  vis  viva  which  is  lost  in  the  impact  by  the  other. 

f  Let  PI  and  P2  represent  resistances  opposed  to  the  motions  of  two  im- 

TTT  -m- 

pinging  bodies  whose  weights  are  Wi  and  W2  ;    also  let  —  -/lf  and    —  -ft  re- 

y  y 

present  the  effective  forces  upon  the  two  bodies  at  any  period  of  the  impact  ; 
then,  by  D'Alembert's  principle, 


or  representing  by  t  the  time  occupied  in  the  impact,  up  to  the  period  of 
greatest  compression,  by  V  their  common  velocity  at  that  period,  and  by  Vi 
and  v2  their  velocities  at  any  period  of  the  impact,  and  substituting  for  fi  and 
/2  their  values  (equation  72), 

Wldvl  W,dv2 

~i~dt~Fl~  Y'd 

Transposing  and  integrating  between  the  limits  0  and  t, 

t 


Now  if  PI  and  P2  be  not  exceedingly  great,  the  integral  in  the  second  member 
of  the  equation  is  exceedingly  small  as  compared  with  its  other  terms,  and  may 
be  neglected  ;  the  above  equation  will  then  become  identical  with  equation 
(700). 


560  IMPACT. 

which  a  nail  will  be  driven  by  the  blow  of  a  hammer  ;  and 
let  it  be  supposed  that  the  resistance  opposed  to  the  driving 
of  the  nail  is  partly  a  constant  resistance  overcome  at  its 
point,  and  partly  a  resistance  opposed  by  the  friction  of  the 
mass  into  which  it  is  driven  upon  its  sides,  varying  in  amount 
directly  with  the  length  of  it  #,  at  any  time  imbedded  in  the 
wood.  Let  this  resistance  be  represented  bya-f/fo;  then 
will  the  work  which  must  be  expended  in  driving  it  to  a 
depth  D  be  represented  (Art.  51.)  by 


J(a+Px)dx,  or  by  (a 


Let  W,  represent  the  weight  of  the  nail,  and  Y  the  velocity 
with  which  a  hammer  whose  weight  is  "W,  must  impinge 
upon  it  to  drive  it  to  this  depth,  and  let  the  surfaces  of  the 
nail  and  hammer  both  be  supposed  inelastic  ;  then  will  the 
work  accumulated  in  the  hammer  before  impact  be  repre- 

W 

sented  by  £  —  'Y2,  and  the  amount  of  this  work  lost  during 

y 
the  impact  by  the  compression  of  the  surfaces  of  contact  will 

1  /  "W  W    \ 

be  represented  (equation  711)  by  —  l^-i-^-JY2.  The  work 

remaining,  and  effective  to  drive  the  nail,  is  therefore  repre- 
sented by  the  difference  of  these  quantities  ;  and  this  work 
being  that  actually  expended  in  driving  the  nail,  we  have 


1  YaW2 

-^     ^.=2aP+^Pa  ....  (712); 

by  the  solution  of  which  quadratic  equation,  D  may  be  deter- 
mined. 


444.  TWO  SOLID  PRISMS  HAVE  A  COMMON  AXIS;  THE  EXTREM- 
ITY OF  ONE  OF  THEM  RESTS  AGAINST  A  FIXED  SURFACE,  AND 
ITS  OPPOSITE  EXTREMITY  RECEIVES  THE  IMPACT,  IN  A  HORI- 
ZONTAL DIRECTION,  OF  THE  OTHER  PRISM  :  IT  IS  REQUIRED 
TO  DETERMINE  THE  COMPRESSION  OF  EACH  PRISM,  THE  LIMITS 
OF  PERFECT  ELASTICITY  NOT  BEING  PASSED  IN  THE  IMPACT. 

Let  "W  represent  the  weight  of  the  impinging  prism,  and 


IMPACT.  561 

Y  its  velocity  before  impact  ;  Lx  and:  L3  the-  lengths  of  the 
prisms  before  compression  ;  Et  and  E2  their  moduli  of  elas- 
ticity ;  Kj  and  K2  their  sections  ;  ^  and  Z2  the  greatest  com- 
pressions produced  in  them  respectively  by  the  impact  ; 
then  will  the  amounts  of  work  which  must  have  been  done 
upon  the  prisms  to  produce  these  compressions  be  repre- 
sented (equation  (486)  by  the  formulae 


_) 

and  the  whole  work  thus  expended  by 


w 

But  this  work  has  been  done  by  the  work  -J — Y*,,  accumu- 
lated (Art.  66)  before  impact  in  the  impinging  body,  and 
that  work  has  been  exhausted  in  doing  it ; 


Moreover,  the  mutual  pressures  upon  the  surfaces  of  con- 
tact are  at  every  period  of  the  impact  equal,  and  at  the 
instant  of  greatest  compression  they  are  represented  respec- 

K  E  I         K  E  I 

tively  (equation  485)  by     t  '  '  and     I  *  2  ; 


Eliminating  Z2  between  this  equation  and  the  preceding,  and 
reducing, 


in  which  expressions  Z,  represents  the  greatest  compression 

36 


562  IMPACT. 

of  the  prism  whose  section  is  K,,  and  P  the  driving  pressure 
at  the  instant  of  greatest  compression. 

445.  The  mutual  pressures  P  of  the  surfaces  of  contact  at 
any  period  of  the  impact. 

Let  I  represent  the  space  described  by  that  extremity  of 
the  impinging  prism,  by  which  it  does  not  impinge :  it  is 
evident  that  this  space  is  made  up  of  the  two  corresponding 
compressions  of  the  surfaces  of  impact  of  the  prisms  ;  so  that 
if  these  be  represented  by  ^  and  199  then  1=^  +  1^  But 

(equation  713)   ^=^-^,4=^;    therefore  1  =  ^*1- 
L, 


446.  A  measure  of  the  compressibility  of  the  prisms. 

If  X  be  taken  to  represent  the  space  through  which  that 
extremity  of  the  impinging  prism  by  which  it  does  not 
impinge  will  have  moved  when  the  mutual  pressure  of  the 
surfaces  of  contact  is  1  Ib.  ;  or,  in  other  words,  if  X  repre- 
sent the  aggregate  space  through  which  the  prisms  would 
be  compressed  by  a  pressure  of  1  Ib.  ;  then,  by  the  preced- 
ing equation, 


^  X  may  be  taken  as  a  measure  of  the  aggregate  compressi- 
bility of  the  prisms,  being  the  space  through  which  their 
opposite  extremities  would  be  made  to  approach  one  another 
by  a  pressure  of  1  Ib.  applied  in  the  direction  of  their 
length. 

If  \  and  \  represent   the   spaces   through    which  the 
prisms  would  severally  be  compressed  by  pressures  of  1  Ib. 

applied  to  each,  then  \=w-^-.  X2—  —  ^-  ;  therefore  X=X  -f 


\,  or  the  aggregate  compressibility  of  the  two  prisms  is 
equal  to  the  sum  of  their  separate  compressibilities. 


IMPACT.  563 


447.  The  work  u  expended  upon  the  compression  of  the 
prisms  at  any  period  of  the  impact. 

The  work  expended  upon  the   compression  Zt  is  repre- 

TT  Tp 

sented  by  -J-  -r—  l^9  ;  or  substituting  its  value  for  ^  (equation 
-U 

713),  it  is  represented  by  ig-g-P2.   And,  similarly,  the  work 
expended  on  the  compression  £,  is  represented  by  8 


therefore  u—  i(f^  +^4^  )P8;    or  substituting  for   P  its 
M^iLj     J\.2ii(a/ 

value  from  equation  (716), 


448.  The  velocity  of  the  impinging  body  at  any  period  of  the 
impact,  the  impact  being  supposed  to  take  place  vertically. 

It  is  evident  that  at  any  period  of  the  impact,  when  the 
velocity  of  the  impinging  body  is  represented  by  -y,  there 
will  have  been  expended,  upon  the  compression  of  the  two 
bodies,  an  amount  of  work  which  is  represented  by  the 
work  accumulated  in  the  impinging  body  before  impact, 
increased  by  the  work  done  upon  it  by  gravity  during  the 
impact,  and  diminished  by  that  which  still  remains  accu- 

W  W 

mulated  in  it,  or  by  £— • Y2+ Wl— \— v\ 

y  y 

Representing,  therefore,  by  u  the  work  expended  upon 

W 

the   compression  of    the  bodies,   we  have  -J — Ya-f"W7— 

\ — ^—u. 

V 

Substituting,  therefore,  for  u  its  value  from  equation 
(718), 


IMPACT. 


Or  substituting  for  I  its  value  in  terms  of  P  (equation  716), 


THE  PILE  DEIVEB. 

449.  It  is  evident  that  the  pile  will  not  begin  to  be 
driven  until  a  period  of  the  impact  is  at- 
tained, when  the  pressure  of  the  ram  upon 
its  head,  together  with  the  weight  of  the 
pile,  exceeds  the  resistance  opposed  to  its 
motion  by  the  coherence  and  the  friction  of 
the  mass  into  which  it  is  driven.  Let  this 
resistance  be  represented  by  P  ;  let  Y  repre- 
sent the  velocity  of  the  ram  at  the  instant  of 
impact,  and  v  its  velocity  at  the  instant  when 
the  pile  begins  to  move,  and  W15  W2  the 
weights  of  the  ram  and  pile  ;  then,  since  the 
pile  will  have  been  at  rest  during  the  whole 
of  the  intervening  period  of  the  impact,  since 
moreover,  the  mutual  pressures  Q  of  the  sur- 
faces of  contact  are  at  the  instant  of  motion 
represented  by  P  —  W2J  we  have  by  equation 
(720) 


If  the  value  of  v  determined  by  this  equation  be  not  a 
possible  quantity,  no  motion  can  be  communicated  to  the 
pile  by  the  impact  of  the  ram  ;  the  following  inequality  is 
therefore  a  condition  necessary  to  the  driving  of  the  pile, 


After  the  pile  has  moved  through  any  given  distance,  one 
portion  of  the  work  accumulated  in  the  ram  before  its 
impact  will  have  been  expended  in  overcoming,  through 
that  distance,  the  resistance  opposed  to  the  motion  of  the 


IMPACT.  565 

pile;  another  portion  will  have  been  expended  upon  the 
compression  of  the  surfaces  of  the  ram  and  pile ;  and  the 
remainder  will  be  accumulated  in  the  moving  masses  of  the 
ram  and  pile.     The  motion  of  the  pile  cannot  cease  until 
after  the  period  of  the  greatest  compression  of  the  ram  and 
pile  is  attained ;  since  the  reaction  of  the  surface  of  the  pile 
upon  the  ram,  and  therefore  the  driving  pressure  upon  the 
pile,  increases   continually  with  the  compression.     If  the 
surfaces  be  inelastic,  having  no  tendency  to  recover  the 
forms  they  may  have  received  at  the  instant  of  greatest 
compression,  they  will  move  on  afterwards  with  a  common 
velocity,  and  come  to  rest  together ;  so  that  the  whole  work 
expended    prejudicially   during  the    impact  will   be   that 
expended  upon  the  compression  of  the  inelastic  surfaces  of 
the  ram  and  pile.     If,  however,  both  surfaces  be  elastic, 
that  of  the  ram  will  return  from  its  position  of  greatest 
compression,  and  the  ram  will  thus  acquire  a  velocity  rela- 
tively to  the  pile,  in  a  direction  opposite  to  the  motion  of 
the  pile.    Until  it  has  thus  reached  the  position  in  respect  to 
the  pile  in  which  it  first  began  to  drive  it,  their  mutual 
reaction  Q  will  exceed  the  resistance  P,  and  the-  pile  will 
continue  to  be  driven.     After  the  ram  has,  in  its  return, 
passed  this  point,  the  pile  will  still  continue  to  be  driven 
through  a  certain  space,  by  the  work  which  has  been  accu- 
mulating in  it  during  the  period  in  which  Q  has  been  in 
excess  of  P.     When  the  motion  of  the  pile  ceases,  the  ram 
on  its  return  will  thus  have  passed  the  point  at  which  it 
first  began  to  drive  the  pile :  if  it  has  not  also  then  passed 
the  point  at  which  its  weight  is  just  balanced  by  the  elas- 
ticity of  the  surfaces,  it  will  have  been  continually  acquiring 
velocity  relatively  to  the  pile  from  the  period  of  greatest 
compression;   it  will  thus  have  a  certain  velocity,  and  a 
certain  amount  of  work  will  be  accumulated  in  it  when  the 
motion  of  the  pile  ceases :    this  amount  of  work,  together 
with  that  which  must  have  been  done  to  produce  that  com- 
pression which  the  surfaces  of  contact  retain  at  that  instant, 
will  in  no  respect  have  contributed  to  the  driving  of  the 
pile,  and  will  have  been  expended  uselessly.     If  the  ram  in 
its  return  has,  at  the  instant  when  the  motion  of  the  pile 
ceases,  passed  the  point  at  which  its  weight  would  just  be 
balanced  by  the  elasticity  of  the  surfaces   of  contact,  its 
velocity  relatively  to  the  pile  will  be  in  the  act  of  diminish- 
ing ;  or  it  may,  for  an  instant,  cease  at  the  instant  when  the 
pile  ceases  to  move.     In  this  last  case,  the  pile  and  ram,  for 
an  instant,  coming  to  rest  together,  the  whole  work  accumu- 


566  IMPACT. 

lated  in  the  impinging  ram  will  have  been  usefully  expended 
in  driving  the  pile,  excepting  only  that  by  which  the  remain- 
ing compression  of  the  surfaces  has  been  produced ;  which 
compression  is  less  than  that  due  to  the  weight  of  the  rara. 
This,  therefore,  may  be  considered  the  case  in  which  a  maxi- 
mum useful  effect  is  produced  by  the  ram.  The  following 
article  contains  an  analytical  discussion  of  these  conditions 
under  their  most  general  form. 


450.  A  prism  impinged  upon  ~by  another  is  moveable  in  the 
direction  of  its  axis,  and  its  motion  is  opposed  by  a  con- 
stant pressure  P ;  it  is  required  to  determine  the  con- 
ditions of  the  motion  during  the  period  of  impact,  the 
circumstances  of  the  impact  being  in  other  respects  the 
same  as  in  Article  448. 

Let  yj  and  f^  represent  the  additional  velocities  which 
would  be  lost  and  acquired  per  second  (see  Art. 
95)  by  the  impinging  prism  and  the  prism 
impinged  upon,  if  the  pressures,  at  any  instant 
operating  upon  them,  were  to  remain  from  that 

W      W 

instant   constant ;   then  will  — -1/.,  — VI  represent 

gj    gtj 

the  effective  forces  upon  the  two  bodies  (Art.  103) 
or  the  pressures  which  would,  by  the  principle  of 
D'Alembert,  be  in  equilibrium  with  the  unbal- 
anced pressures  upon  them,  if  applied  in  opposite 
directions. 

Now  the  unbalanced  pressure  upon  the  system 
BP  composed  of  the  two  prisms  is  represented  by 

(W1+W2-P), 

"W1        W2 

"~g~       y*    ~    *       2~~ 

also  the  unbalanced  pressure  upon  the  prism  PQ=W2-f- 
Q— P,  where  Q  represents  the  mutual  pressure  of  the  prisms 
atQ; 


Let  A  have  been  the  position  of  the  extremity  B  of  the 
impinging  prism  at  the  instant  of  impact ;  and  let  xl  repre- 
eent  the  space  through  which  the  aggregate  length  BP  of 
the  two  prisms  has  been  diminished  since  that  period  of  the 


IMPACT.  567 

impact,  and  a?3  the  space  through  which  the  point  P  has 
moved  ;  then  (equation  716) 

L'         L»       1^ 


Also  AB =2^+0?,,;  therefore  velocity  of  point  B= 
(Art.  96);  therefore  f*=-j-++-j-£=-j-+- 


\MV  WV  WV 

Substituting  these  values  of  ft  and  Q  in  equations  (723)  and 
(724),  and  eliminating/^  between  the  resulting  equations, 


Integrating  this  equation  by  the  known  rules,t  we  obtain 
aj^A  sin.  /£+B  cos.  7^  +  -r-  .....  (727); 


in  which  expression  the  value  of  7  is  determined  by  the 
equation 

^(W+WJ  =9  |  L/Kl^+Sk)-'  f  '  '  '  ' 


and  A  and  B  are  certain  constants  to  be  determined  by  the 
conditions  of  the  question.  Substituting  in  equation  (724) 
the  value  of  Q  from  equation  (725),  and  solving  in  respect 
to/* 


(729). 


Substituting  for  xl  its  value  from  equation  (727),  and  for  ft 
its  value  -r^,  and  reducing, 

<#X      Ag    .  B<7 

^=w?sin-7*+w>cos- 

Integrating  between  the  limits  0  and  £,  and  observing  that 

yy/v» 

when  tf=0,  -77=0 ;  the  time  being  supposed  to  commence 

CM 

with  the  motion  of  the  prism  PQ, 

*  Art.  96.     Equations  (72)  and  (74). 
f  Church's  Int.  Cal.     Art.  183. 


568  IMPACT. 


sn- 


Integrating  a  second  time  between  the  same  limits, 

*•  =  vw  ^  ~  8in*  7f}  +        ~  cos* 


Now  when  the  motion  of  the  second  prism  ceases  -^7=0  ; 

d/t 

whence,  if  the  corresponding  value  of  t  be  represented  by  T, 
A(l-cos.  yT)+B8in.  7  T  +    1- 


To  determine  the  constants  A  and  B,  let  it  be  observed 
that  the  motion  of  the  prism  QP  cannot  commence  until  the 
pressure  Q  of  the  impinging  prism  upon  it,  added  to  its  own 
weight  W2,  is  equal  to  the  resistance  r  opposed  to  its  motion. 
So  that  if  c  be  taken  to  represent  the  value  of  xl  (i.  e.  the 
aggregate  compression  of  the  two  prisms)  at  that  instant, 

then,  substituting  for  Q  its  value  from  equation  (725),  -  + 

X 

W,=P; 

....  (732). 


Now  since  the  time  t  is  supposed  to  commence  at  the 
instant  when  this  compression  is  attained,  and  the  prism  PQ 
is  upon  the  point  of  moving,  substituting  the  above  value  of 
c  for  xl  in  equation  (727),  and  observing  that  when  x=c, 

t=Q,  we  obtain  (P—  Wa)X=:B+    3^.  ;  whence  by  substitu- 

7  "s 
tion  from  equation  (728),  and  reduction, 


So  long  as  the  extremity  P,  of  the  prism  impinged  upon, 
is  at  rest,  the  whole  motion  of  the  point  B  arises  from  the 

compression  of  the  two  prisms,  and  is  represented  by  -^. 


IMPACT.  569 

The  value  of       *  ?  when  t=0,  is  represented  therefore  by  v 

dt 
(equation  721).     Differentiating,  therefore,  equation  (727), 

assuming  £=0,  and  substituting  v  for-^-1;  we  obtain  v=yA; 

whence  it  appears  that  the  value  of  A  is  determined  by 
dividing  the  square  root  of  the  second  number  of  equation 
(721)  by  r 

Substituting  for  A  and  B  their  values  in  equations  (731-3) 


-lsin.rT+ 


Reducing,  and  dividing  by  the  common  factor  of  the  two 
last  terms, 


Substituting  for  A  and  B  their  values  in  equation  (730),  and 
representing  by  D  the  value  of  a?a,  when  £=T, 


....  (735). 

The  value  of  T  determined  by  equation  (734)  being  sub- 
stituted in  equation  (735),  an  expression  is  obtained  for  the 
whole  space  through  which  the  second  prism  is  driven  by 
the  impact  of  the  first.* 

*  The  method  of  the  above  investigation  is,  from  equation  (726),  nearly  the 
same  with  that  given  by  Dr.  Whewell,  in  the  last  edition  of  his  Mechanics*;  the 
principle  of  the  investigation  appears  to  be  due  to  Mr.  Airey.  If  the  value 
of  y,  as  determined  by  equation  (734),  were  not  exceedingly  great,  then,  since 
the  value  of  T  is  in  all  practical  cases  exceedingly  small,  the  value  of  yT  would 
in  all  cases  be  exceedingly  small,  and  we  might  approximate  to  the  value  of 
T  in  equation  (735),  by  substituting  for  cos.  yT  and  sin.  yT,  the  two  first  terms 
of  the  expansions  of  those  functions,  in  terms  of  yT. 


EDITORIAL  APPENDIX. 


NOTE  (a). 

BESIDES  its  direction  defined  (Art.  1),  we  have  also  to  take 
into  consideration,  in  estimating  the  effects  of  a  force,  its 
point  of  application,  or  the  point  of  the  body  where  it  acts, 
either  directly  or  through  the  medium  of  some  other  body, 
as  a  rigid  bar,  or  an  inextensible  cord  in  its  line  of  direction ; 
the  point  on  its  line  of  direction  towards  which  the  point  of 
application  has  a  tendency  to  move ;  and  finally  the  inten- 
sity, or  magnitude  of  the  force  as  expressed  in  terms  of  some 
settled  unit  of  measure. 

NOTE  (b\ 

This  result  of  experiment  also  admits  of  the  following 
proof:     Let  A  be  the  point  of  appli- 

•* — «•* *>     >•» — «<P    cation  of  a  force  P,  and  let  this  point 

be  invariably  connected  with  another 

point  B,  in  the  line  of  direction  towards  which  A  tends  to 
move  from  the  action  of  P ;  suppose  now  two  other  forces 
P,  and  P2,  each  equal  to  P,  to  be  applied  ;  the  one  at  A,  in 
a  direction  opposite  to  P,  and  the  other  at  B,  in  the  same 
direction  as  P ;  the  introduction  of  these  two  equal  forces, 
acting  in  opposite  directions,will  evidently  in  no  wise  change 
the  direction  or  intensity  of  P ;  but  as  Pa  is  equal  and  oppo- 
site to  P  its  eifect  will  be  to  balance  the  action  of  P  at  A, 
whilst  it  leaves  P2  to  exert  an  action  at  B  precisely  the  same 
as  P  was  exerting  at  A  before  the  introduction  of  J?,  and  Pa. 


NOTE  (c). 
Suppose  two  forces  P1  and  Pa  applied  to  the  same  point  A, 

571 


572  EDITORIAL   APPENDIX. 

the  direction  of  the  one  being  AB,  that  of  the 
other  AC ;  no  was  these  forces  make  an  angle 
with  each  other,  it  is  evident,  as  the  point  of 
application  can  move  but  in  one  direction,  and 
as  it  is  solicited  to  move  towards  B  and  C  at 
the  same  time,  that  it  must  move  in  some 
direction  which  is  coincident  with  neither  of 
these;  this  direction,  it  is  equally  evident, 
must  be  in  the  same  plane  as  the  directions  AB  and  AC,  for 
there  is  no  argument  in  favor  of  a  direction  assumed  exterior 
to  the  plane  and  on  one  side  of  it  which  would  not  equally 
apply  to  a  symmetrical  direction  assumed  on  the  other  side ; 
it  is  also  evident  that  this  direction  must  be  some  one  AF 
within  the  angle  formed  by  AB  and  AC,  for  the  point,  if 
solicited  by  Pj  alone,  would  take  the  direction  AB,  and  as  it 
cannot  take  a  direction  to  the  left  of  BD,  as  there  is  no  force 
that  solicits  it  on  that  side,  and,  for  like  reasons,  cannot  take 
one  to  the  right  of  CE,  it  must  therefore  take  the  one 
assigned  somewhere  within  the  angle  BAC. 

]Now  suppose  further  that  Px  and  P3  are  equal,  it  is  evi- 
dent that  the  direction  assigned  to  their  resultant,  or  that  of 
the  motion  of  their  point  of  application,  must  be  the  one 
which  bisects  the  angle  BAC,  for  the  argument  for  any 
direction  on  the  left  of  this  line  would  be  equally  cogent  for 
the  like  position  on  the  other  side. 

If  Pj  and  P2  are  unequal  then  will  the  direction  of  their 
resultant  divide  the  angle  BAC  unequally,  the 
smaller  portion  being  next  to  the  greater  force  ; 
for  suppose  Pj  divided  into  two  portions,  one 
of  which  P  shall  be  equal  to  P2 ;  P  and  P2  can 
be  replaced  by  their  resultant  Rj,  the  direction 
of  which  AF  bisects  the  angle  BAC ;  we  shall 
then  have  two  forces  Kt  and  the  remaining 
portion  of  P1?  the  resultant  of  which  R  must  lie 
somewhere  within  the  angle  BAF,  and  there- 
fore nearer  to  Px  than  to  P2 ;  but  It  is  the  resultant  of  the 
two  forces  Pl  and  P2.  Therefore,  &c. 

Hence  it  is  seen  that  two  forces  wrhose  directions  form  an 
angle  between  them  and  meet,  1st,  have  a  resultant ;  2nd,  that 
the  direction  of  this  resultant  lies  in  the  plane  of  the  two 
forces  ;  3d,  that  it  passes  through  the  point  where  the  direc- 
tions meet,  and  lies  within  the  angle  contained  between 
them ;  4th,  that  it  bisects  this  angle  when  the  forces  are 
equal ;  5th,  that  when  the  forces  are  unequal  it  divides  this 
angle  unequally,  the  smaller  portion  being  next  to  the  greater 
force. 


EDITOKIAL   APPENDIX. 


Now  as  the  two  forces  Px  and  P2  can  be  replaced  by  their 
resultant  R,  and  as  the  effect  of  this  will  be  the 
same  if  applied  at  any  point  F  in  its  line  of 
direction  as  at  the  point  of  application  of  the 
two  forces,  it  is  evident,  if  we  transfer  P,  and  P7 
also  to  F,  preserving  their  new  parallel  to  theii 
original  directions,  that  they,  in  turn,  can  be  made 
to  replace  R.  It  thus  appears  that  the  point  of 
application  of  two  forces  may  be  transferred  to 
any  point  of  the  line  of  direction  of  their  result- 
ant without  changing  the  effects  of  these  forces,  provided 
their  new  directions  are  kept  parallel  to  their  original  ones. 
It  is  upon  the  preceding  propositions,  in  themselves  self- 
evident,  that  the  mode  of  demonstration,  known  as  Duchay- 
la's,  of  the  proposition,  termed  the  parallelogram  of  forces, 
or  of  pressures,  is  based. 

NOTE  (d). 

When  two  parallel  forces  are  applied  to  two  points  inva- 
riably connected,  their  resultant  can  be  found  by  applying 
the  propositions  in  (Arts.  1,  2,  3). 

Let  P,  and  P2  be  two  parallel  forces  applied  at  the  points 
A  and  B  invariably  connected,  as  by  a 
rigid  bar.  Let  two  equal  forces  Qx  and 
Q2  be  so  applied,  the  one  at  A  the  other 
at  B,  as  to  act  in  opposite  directions 
along  AB.  These  two  will  evidently 
have  no  effect  to  change  the  action  of 
P,  and  P2.  Now  the  two  forces  P,  and 
Q,  applied  at  A  will  have  a  resultant  R1? 
the  intensity  and  direction  of  which  can 
be  found  by  the  preceding  method.  In  like  manner  the 
resultant  R2  of  P2  and  Q2  can  be  obtained.  Now  the  forces 
being  replaced  by  their  resultants,  the  equilibrium  will  still 
subsist,  and  the  effect  will  remain  the  same  whether  Rx  and 
R2  act  at  A  and  B,  or  at  o  their  point  of  meeting.  But  as 
Rt  and  R2  can  each  be  replaced  by  their  components  at  any 
point  of  their  direction,  let  these  components  be  transferred 
to  the  point  o.  In  this  position  Qx  and  Q2  will  destroy  each 
other,  whilst  P,  and  P3  will  act  in  the  same  direction  along 
<?C  and  parallel  to  their  original  ones,  with  an  intensity  equal 
to  their  sum  Pj-f  P2. 

Now  from  the  similar  triangles  A#C,  rom  /  and  B0C,  son, 
there  obtains, 


574:  EDITORIAL   APPENDIX. 

om  :  mr  : :  oC  :  CA,  or  Px :  Q,  : :  oC  :  CA. 
ns  :  on  ::  CB  :  00,  or  Q,  :  P2  ::  CB  :  00. 

Multiplying  the  two  last  proportions,  there  obtains, 

I\  :  P2 : :  CB  :  CA, 
and 

P, :  Pa :  P,+  Pa ::  CB  :  CA  :  CB+CA  or  AB. 

From  this  we  see  that  two  parallel  forces  acting  in  the 
same  direction,  1st,  have  a  resultant  which  is  equal  to  their 
sum ;  2nd,  that  the  direction  of  this  resultant  is  parallel  to 
that  of  the  forces;  3d,  that  it  divides  the  line  joining  the 
points  of  application  of  the  two  forces  into  parts  reciprocally 
proportional  to  the  forces ;  4th,  that  either  force  is  to  the 
resultant  as  the  portion  of  the  line  between  the  resultant  and 
the  other  force  is  to  the  total  distance  between  the  points  of 
application ;  5th,  that  the  foregoing  propositions  hold  true 
for  any  position  of  the  line  AB  with  respect  to  the  two 
parallel  forces  and  their  resultant. 

When  the  two  forces  act  in  opposite  directions  at  the 

points  A  and  B,  by  following 

\:;;:f*    a  like  process,  we  obtain  the 

„--••"'"*  two  resultants  Ex   and  E2, 

...--"'"  which    being  prolonged  to 

their  point  of  meeting  o  we 
can  again  replace  them  by 

/  /  \  their  components  P1?  Qx,  and 

klJ  P2,  Q2 ;  of  which  Pt  and  P2, 

acting  parallel  to  their  ori- 
ginal positions  but  in  opposite  directions,  will  have  for  their 
resultant  P,— P2. 

Now  prolonging  the  direction  of  this  resultant  until  it 
meets  AB  prolonged  at  C,  there  obtains  as  in  the  preceding 
case,  from  the  similar  triangles  A0C,  rom*  and  B0C,  son, 

om  :  mr  : :  0C  :  CA,  or  Pa  :  Q,  : :  00  :  CA, 
ns  :  on  ::  CB  :  00,  or  Q3 :  P, ::  CB  :  0C, 
hence, 

P, :  Pa :  P.-P, ::  CB  :  CA  :  CB-CA  or  AB. 


EDITORIAL    APPENDIX. 


575 


Remark. — Although  it  may  be  assumed,  as  self-evident, 
that  any  resultant  can  be  replaced 
by  two  equivalent  components, 
without  disturbing  the  equilibrium, 
and  that  each  of  these  in  turn 
may  be  replaced  by  two  other 
equivalent  components,  and  so  on 
for  any  number  of  components ;  still  like  compositions  and 
resolutions  of  forces  are  of  such  frequent  occurrence  in  esti- 
mating the  pressures,  or  strains  on  the  various  points  of  any 
mechanical  contrivance,  as  a  machine,  a  frame  of  timber, 
&c.,  arising  from  a  resultant  pressure,  that  the  student  can- 
not be  too  familiar  with  the  processes  of  effecting  such  com- 
positions and  resolutions. 

To  show  by  a  simple  illustration  this  truth,  let  the  result- 
ant AR  be  replaced  by  its  two  equivalent  components  AP, 
and  AP2  in  any  assumed  positions ;  and  let  each  of  these 
components  be  resolved  into  two  others,  AQ15  AE^  for  APj ; 
and  AQ2,  AR2  for  AP2,  taken  respectively  perpendicular  and 
parallel  to  AR.  Now  it  is  evident,  from  the  figure,  that  the 
two  components  AQ1?  AQ2  of  this  last  resolution  are  equal 
and  opposite  in  direction,  and  therefore  destroy  each  other ; 
whilst  the  two  AR15  AR2  act  in  the  direction  of  AH,  and 
their  sum  is  equal  to  AH.  The  same  may  in  like  manner  be 
shown  for  any  number  of  sets  of  components  by  which  AH 
might  be  replaced. 


NOTE  (e). 

If  the  point  o  from  which  perpendiculars  are  drawn  to  the 
directions  of  two  forces  P:  and  P2,  is 
taken  on  the  direction  of  their  re- 
sultant, then  will  mtP1=mtP2. 

For  from  o  draw  the  perpendicu- 
lars om,  on,  to  Pa  and  "r,,  and  join 
the  points  m  and  n  of  their  inter- 
section. The  quadrilateral  A.mon, 
having  the  angles  at  m  and  n  right 
angles,  can  be  inscribed  in  a  circle,  therefore  the  two  angles 
at  m  and  A,  subtended  by  the  chord  on,  will  be  equal.  In 
the  triangles  mon  and  ABC,  the  angle  o  is  the  supplement 
of  the  angle  A  of  the  quadrilateral,  and  B,  being  the  adja- 
cent angle  of  the  parallelogram,  is  also  the  supplement  of  A ; 


576  EDITORIAL   APPENDIX. 

the  two  triangles,  having  two  angles  equal,  are  similar, 
therefore, 

AB  :  BC  :  :  om  :  on,  or  P3  :  P:  :  :  om  :  on  ; 

hence 

=~Pi  x  on.    Therefore,  &c. 


From  this  proposition  the  relations  of  two  parallel  forces 
to  their  resultant  can  be  readily  deduced  from  the  limiting 
case  of  the  angle  mon  of  the  triangle  ;  for  from  the  two  simi- 
lar triangles  there  obtains  as  before 

P3  :  P1  :  R  or  AC  :  :  om  :  on  :  mn. 

Now  as  this  is  true  for  any  value  of  the  angle  0,  when  it 
becomes  180°,  the  forces  P,,  P2  having  the  same  direction, 
and  their  resultant  E  become  parallel;  the  perpendiculars 
om  and  on  become  portions  of  the  line  mn  ;  and,  as  mn=om 
+  on,  it  follows,  from  the  above  proportion,  that  R—  Pj  +  P,. 

When  Pj  and  P2  have  opposite  directions,  we  can  suppose 
the  force  Pa,  for  example,  and  its  perpendicular  turned  about 
the  point  o  in  the  plane  of  the  forces  until  the  point  m  falls 
on  the  prolongation  of  on  on  the  opposite  side  from  0,  in 
which  position  Px  and  P2  again  become  parallel,  but  act  in 
opposite  directions.  During  this  rotation  of  P1?  the  resultant 
still  passes  through  o,  and  there  still  obtains 

Pa  :  P!  :  :  R  :  om  :  on  :  mn  ; 

but,  as  mn  now  is  equal  to  om—on,  it  follows,  from  the 
proportion,  that  R=Pa—  Pr  Hence  the  same  relations 
between  P,,  Pa  and  R  as  already  established,  NOTE  (d). 


^  Otherwise,  since  in  any  number  of  forces  in  equilibrium 
either  of  them  is  equal  and  directly  opposed  to  the  resultant 
of  all  the  rest,  the  whole  may  be  replaced  by  these  two 
without  disturbing  the  equilibrium.  If  now  through  the 
point  of  application  of  these  two  we  draw  any  two  lines  at 
right  angles  to  each  other,  and  then  resolve  each  of  the  two 
forces  into  two  components  parallel  to  these  two  lines,  it 
will  be  at  once  seen,  from  the  diagram,  that  the  like  com- 


EDITORIAL   APPENDIX.  577 

portents  will  be  equal  and  opposite  to  each  other,  and  this 
would  evidently  be  the  same  for  the  components  of  the  ori- 
ginal forces  resolved  in  the  same  manner,  otherwise  there 
would  be  a  resultant  for  all  the  forces,  which  is  contrary  to 
the  supposition  of  an  equilibrium. 

.Remark. — As  this  method  of  resolving  a  system  of  forces 
into  sets  of  components  parallel  to  any  assumed  rectangular 
axes,  in  order  to  determine  their  algebraical  values,  is  of 
frequent  use,  in  simplifying  the  numerical  calculations 
necessary  in  the  applications  of  the  principles  of  statics,  the 
student  should  make  himself  perfectly  familiar  with  the  pro- 
positions that  precede  and  follow  Art.  11. 


(g). 

Otherwise,  join  DE  which  will  be  parallel  to  AC^thus 
forming  with  it  and  the  lines  AD  and  CE  two  equi-angular 
triangles,  from  which  there  obtains 

DE:DG::  AC:  AG; 
but  DE=f  AC,  therefore  DG=4AG=£DA. 

NOTE  (A). 

Otherwise,  join  GH  which,  as  AG  and  CH  intersect,  will 
be  in  the  same  plane  with  them  and  with  the  line  AC.  As 
AH  and  CG  are  respectively  -f  of  the  lines  drawn  from  A 
and  C  to  the  middle  of  BD,  it  follows  that  GH  is  parallel  to 
AC  and  forms  with  it  and  the  lines  AG  and  CH,  by  their 
intersection  at  K,  the  two  equi-angular  triangles  CKH  and 
and  AKC,  from-  which  there  obtains 

GH:GK::  AC  :  AK, 
but  GH=|AC,  therefore  GK=£  AK=iAG. 


As  the  methods  employed  in  (Art.  45,  &c.)  to  represent, 
by  geometrical  diagrams,  what  are  termed  the  laws  of 
motion,  or  the  relations  which  exist  at  any  two  given 

37 


m 

-V 


578  EDITORIAL   APPENDIX. 

instants  between  the  velocity,  the  space,  and  the  time  of  a 
body's  motion,   although  very  simple   in   themselves,   are 
sometimes  found  to  offer  difficulties  to  the  student,  particu- 
larly as  to  the  representation  of  spaces  by  areas,  a  few  addi- 
tional marks  on  this  point  may  not  be  here  misplaced. 
Taking,  in  the  first  place,  the  case  of  a  body  M  moving  in 
D  a  rectilinear  path  from  A  towards 

IB  with  a  uniform  motion.  Ac- 
cording to  the  definition,  the 
body  will  move  over  the  suc- 

5.  cessive  equal  portions  Ab,  be,  cd, 

&c.,  of  its  path  in  equal  succes- 
sive portions  of  time,  however  small  or  great  these  portions 
may  be.   Taking  now  any  portion  of  time  as  a  unit,  as  a  second, 
a  minute,  &c.,  and  supposing  Ab  the  portion  of  its  path,  or 
the  space  through  which  the  body  has  moved  during  this 
unit,  Ab  will  represent  what  is  termed  the  velocity,  or  rate 
of  motion  of  the  body ;  and  when  the  path  itself  is  expressed 
in  terms  of  any  linear  unit,  as  a  foot,  a  yard,  a  mile,  &c.,  the 
number  of  these  units  in  Ab  will  measure  the  velocity ;  for 
example,  if  the  unit  of  path,  or  space  is  a  foot,  and  there 
were  four  of  these  units  in  Ab,  and  the  unit  of  time  is  a 
second,  then  the  velocity  would  be  termed  a  velocity  of  four 
feet  per  second,  &c.     Supposing  the  body  to  start  from  A, 
with  this  velocity,  it  will  successively  move  over  distances, 
each  of  four  feet  in  length,  along  its  path,  in  successive 
seconds  of  time  ;  consequently  any  distance,  or  space,  as  Ad, 
will  be  equal  to  Ab  taken  as  many  times  as  the  number  of 
seconds  elapsed  from  the  time  the  body  started  from  A  until 
it  reached  d\  or,  in  other  words,  the  number  of  units  in  the 
space  Ad  is  expressed  by  the  abstract  number  obtained  by 
multiplying  the  number  of  units  in  the  velocity  by  the  num- 
ber of  units  in  the  time.     This,  like  all  other  similar  pro- 
ducts, can  be  expressed  algebraically,  or  geometrically  ;  but 
by  whatever  symbol  expressed,  the  signification  is  the  same. 
For  example,  on  any  two  lines,  as  AB  and  AC,  taken  at 
right  angles,  set  off  any  number  of  equal  parts  as  Ab,  be, 
cd,  &c.,  as  units  of  time,  and  on  AC  any  number  also  of 
equal,  parts,  which  may  be  the  same  in  length,  or  otherwise, 
as  those  on  AB,  to  represent  the  units  in  wnich  the  velocity 
is  expressed.     Suppose  the  latter  to  be  composed  of  the  four 
units  Am,  mn,  &c. ;  and  that  the  number  of  units  of  time 
considered  is  three  ;    on  the  lines  Ad,  AC  construct  the 
rectangle  AD;  then  is  the  area  of  the  rectangle  said  to 
express  the  space  corresponding  to  the  velocity  and  time 


EDITORIAL    APPENDIX.  579 

here  assumed ;  that  is,  the  number  of  units  in  area  of  this 
rectangle,  expressed  in  terms  of  the  unit  of  area  on  Ab  and 
Am  for  example,  is  equal  to  the  number  of  units  of  space. 
In  like  manner  the  area  of  the  rectangle  AE  expresses  the 
space  corresponding  to  the  velocity  and  the  time  Ae,  &c. 
In  uniformly  varied  motion,  as  the  velocity  increases  in 

the  same  proportion  as  the  time 
increases,  or,  in  other  words,  as 
the  augmentations  of  the  velocity 
for  equal  intervals  of  time  is  the 
same,  these  relations  between  the 
B  times,  velocities  and  spaces,  can, 
in  like  manner,  be  expressed  by 

a  geometrical  diagram  as  follows  :  On  any  line,  as  AB,  set 
oft  a  number  of  equal  parts  as  Ab,  be,  ed,  &c.,  to  represent 
equal  intervals  of  time ;  at  the  points  J,  c,  d,  &c.,  having 
drawn  perpendiculars  to  AB,  set  off  on  them  distances  b?n, 
en,  do,  &c.,  to  represent  the  corresponding  velocities ;  in 
which  cn—%bm  do—3bm ;  or  Ad  :  Ac  :  Ab  ::  do  :  en  :  bm, 
&c.  Now,  as  the  same  relations  obtain  between  all  the  dis- 
tances set  off  on  AB  and  their  corresponding  perpendicu- 
lars, it  follows  that  the  line  AC,  drawn  through  the  points 
in,  n,  o,  &c.,  is  a  right  line,  and  that  the  triangles  Abm,  Aon, 
&c.,  are  therefore  similar.  As  the  relations  between  the 
times  and  velocities  are  true,  however  great,  or  however 
small  the  equal  portions  of  time  may  be  assumed,  let  us  sup- 
pose these  portions,  as  Ab,  "be,  cd,  to  be  taken  so  small  that 
the  velocity  of  the  body  during  any  one  of  them  may  be 
considered  uniform,  and  as  a  mean  between  what  it  actually 
is  at  the  commencement  and  end  of  this  portion ;  tkat  is  en 
and  do,  for  example,  representing  the  actual  velocities  at  the 
beginning  and  end  of  the  interval  of  time  represented  by  cd, 
then  J  (en  +  do)  represents^  the  mean,  or  uniform  velocity 
during  this  interval.  This  being  premised,  the  number  of 
units  of  space  over  which  the  body  will  pass  whilst  moving 
with  a  uniform  velocity,  expressed  by  %  (cn  +  do),  during  the 
interval  cd,  will  be  represented,  according  to  the  preceding 
proposition,  by  cdx%  (cn  +  do),  but  this  also  expresses  the 
area  of  the  trapezoid  cdno ;  and  as  the  same  is  true  for  all 
the  like  trapezoids  it  will  also  be  true  for  their  sums,  or  for 
the  triangles,  as  Ado  and  Afq  for  example,  the  areas  of  which 
are  equal  to  the  sum  of  the  areas  of  the  trapezoids  of  which 
they  are  composed.  Supposing  the  body  to  move  from  a 
state  of  rest  with  a  uniformly  accelerated  motion,  and  that 
at  the  intervals  of  time,  represented  by  Ad  and  Af,  its 


580  EDITORIAL   APPENDIX. 

respective  velocities  are  do  andjfy,  then  will  tlie  number  of 
units  of  space  which  the  body  will  have  moved  over  in  these 
two  intervals  be  respectively  expressed  by  the  number  of 
units  of  area  in  the  triangles  Ado  and  Afq.  As  the  trian- 
gles are  similar  their  areas  are  as  the  squares  of  their  like 
sides  ;  it  therefore  follows  that  in  uniformly  varied  motion, 
the  spaces  are  as  the  squares  of  the  times,  or  as  the  squares 
of  the  velocities. 

As  do  represents  the  velocity  acquired  during  the  time 
A<#,  supposing  the  body  to  have  moved  from  a  state  of  rest, 
and  the  number  of  units  of  area  in  the  triangle  Ado  repre- 
sents the  corresponding  number  of  units  of  space,  it  follows, 
that  if  the  body  had  moved,  during  the  same  interval,  with 
the  velocity  do  which  it  actually  acquired  in  it,  the  number 
of  units  of  space  it  would  then  have  passed  overVould  have 
been  represented  by  the  number  of  units  of  area  in  the  rect- 
angle A0,  constructed  on  Ad  and  do.  But,  as  the  area  of 
the  rectangle  is  double  that  of  the  triangle,  the  space  that 
would  have  been  passed  over  in  the  supposed  case  would 
have  been  double  that  passed  over  in  the  actual  case. 

If  we  take  any  portion,  as  A0,  to  represent  the  unit  of 
time,  then  the  corresponding  perpendicular  ep  will  represent 
the  velocity,  or  the  quantity  f  used  in  (Arts.  46.  47)  fol- 
lowing. 

NOTE  (j). 

As  the  propositions  under  this  head,  and  those  under  the 
heads  of  Accumulation  of  Work  in  a  Moving  Body  (Art.  64) 
and  Principle  of  Yis  Yiva  (Art.  129)  constitute  the  basis  of 
what  may  be  termed  Industrial  Mechanics,  or  the  applica- 
tions of  the  principles  of  abstract  mechanics  to  the  calcula- 
tion of  the  effects  of  motive  power  transmitted  by  machines 
and  employed  to  produce  some  'useful  mechanical  end,  it  is 
very  important  that  the  student  should  have  a  clear  and 
definite  apprehension  of  their  signification  in  this  point  of 
view.  Work,  as  here  defined,  supposes  two  conditions  as 
essential  to  its  production  :  a  continued  resistance,  or  obstacle 
removed  by  the  action  of  a  force,  and  a  motion  of  the  point 
of  application  of  the  force  in  a  direction  opposite  to  that  in 
which  the  resistance  acts.  Its  measure  is  expressed  by  the 
product  arising  from  multiplying  the  number  of  units  of  the 
resistance,  or  of  its  equivalent  force  directly  opposed  to  it, 
by  the  number  of  units  of  path  which  the  point  of  applica- 
tion of  this  force  has  described  during  the  interval  consi- 


EDITORIAL   APPENDIX.  581 

dered,  in  which  the  force  acts  to  overcome  the  resistance. 
It  follows  that  the  work  will  be  0  when  this  .product  is  0 ; 
that  is,  when  either  of  the  factors,  the  resistance,  or  its  equi- 
valent force,  or  the  path  described,  is  0. 

In  estimating  work,  that  which  is  external  and  which  alone 
generally  we  have  the  means  of  measuring,  is  alone  consi- 
dered. For  example,  if  with  a  flexible  bar  a  person  attempts 
to  push  before  him  any  obstacle,  the  first  effect  observed  will 
be  a  certain  deflection  of  the  bar,  during  which  the  hand,  at 
one  end  of  the  bar,  .will  have  moved  forward  a  certain  dis- 
tance in  the  direction  of  the  point  of  application  at  the  other, 
producing  an  amount  of  work  which  is  expressed  by  the 
product  of  the  pressure,  or  force  exerted  by  the  hand,  sup- 
posing this  pressure  to  remain  constant  during  this  period, 
and  the  path  described,  in  the  line  of  direction  of  this  pres- 
sure, by  the  point  where  it  is  applied.  During  this  period, 
as  the  obstacle  to  be  moved  has  remained  at  rest,  no  path 
has  been  described  by  the  point  where  the  bar  rests  against 
it,  therefore,  according  to  our  definition,  no  work  has  been 
done  upon  the  resistance.  The  effect  produced  by  the 
pressure  has  been  simply  to  bend  the  bar,  and  the  work  is 
therefore  due  only  to  the  resistance  offered  by  the  molecular 
forces  of  the  material  composing  the  bar  to  the  force  that 
tends  to  bend  it.  This  portion  of  the  work,  although  in  this 
case  we  have  the  means  of  measuring  it,  being  what  may  be 
termed  internal,  is  not  taken  into  the  account  in  estimating 
that  due  to  the  resistance  to  be  overcome,  which  would  have 
been  the  same  had  a  perfectly  rigid  bar  been  used  instead  of 
the  flexible  one. 

In  like  manner,  when  an  animal  carries  a  burthen  on  his 
back  from  one  point  to  another  on  a  horizontal  plane  no 
work  is  produced  according  to  our  definition ;  for  no  resist- 
ance has  been  overcome  in  the  direction  in  which  the  bur- 
then has  been  carried,  and  therefore  the  product  that  repre- 
sents the  work  is  0.  The  work  in  this  case,  as  in  that  of  the 
flexible  bar,  is  internal ;  and  similar  to  that  arising  from  a 
burthen  borne  by  an  animal  whilst  standing  still ;  and  there- 
fore although  both  of  them  may  be  very  useful  operations 
and  have  a  marketable  value,  still  they  can  neither  be  mea- 
sured by  the  standard  by  which  it  is  agreed  to  estimate 
work. 

Every  mechanical  operation  performed  by  machinery  pre- 
sents a  case  of  work.  Take  for  example  the  simple  opera- 
tion of  planing,  in  which  the  hand  moves  a  plane,  which  is 
but  a  rigid  bar  to  which  is  fixed  an  iron  tool  like  a  chisel  for 


582  EDITORIAL   APPENDIX. 

removing  successive  thin  portions  from  the  edge,  or  surface 
of  a  board.  In  this  case  the  resistance  offered,  and  which  is 
sensibly  in  the  same  direction  as  the  power  applied,  is  that 
arising  from  the  cohesion  of  the  fibres  of  the  material,  and 
is  measured  by  the  pressure  applied ;  the  path  which  the 
point  of  application  of  the  iron  tool  describes  is  the  same  as 
that  described  by  the  hand ;  and  the  work  will  be  expressed 
by  the  product  of  these  two  elements,  each  estimated  in 
terms  of  its  own  unit  of  measure.  The  case  of  the  common 
grindstone  presents  an  example  of  a  rather  more  complicated 
character.  Here  the  instrument  to  be  ground  is  pressed 
against  the  periphery  of  the  stone  with  sufficient  force  to 
cause  a  certain  resistance  to  any  power  however  applied  to 
put  the  stone  in  motion.  The  direction  however  in  which 
this  resistance  acts  at  the  point  of  application  is  in  the 
direction  of  the  tangent  to  the  periphery  at  this  point,  and, 
in  one  revolution  of  the  stone,  it  will  describe  a  path  equal 
in  length  to  the  circle  described  by  the  point  of  application. 
The  work  therefore  for  each  revolution  will  be  the  product 
of  the  resistance,  estimated  in  the  direction  of  the  tangent, 
and  the  circumference  described  by  the  point  of  application. 
It  may  be  as  well  to  remark,  in  this  place,  that  although 
the  work  done  in  overcoming  the  molecular  resistances  of 
the  materials  by  means  of  which  the  action  of  a  force  or 
pressure  is  transmitted,  as  in  the  example  above  cited  of  a 
flexible  bar,  is  not  taken  into  account  in  estimating  the 
external  work,  there  are  cases  in  which  this  work  constitutes 
the  entire  work  done,  and  which  again  is  reproduced  in 
external  work ;  as  for  example  in  the  cases  of  the  common 
bow  used  for  projecting  arrows,  and  the  springs  by  which 
the  machinery  of  some  time-pieces  is  moved.  In  each  of 
these  the  resistance  offered  by  the  molecular  forces  of  the 
material  is  overcome  by  the  action  of  some  external  force, 
whose  point  of  application  is  made  to  describe  a  given  path  ; 
by  this  action  a  certain  amount  of  work  is  expended  in 
bringing  the  spring  to  a  certain  degree  of  tension  which, 
when  the  force  is  withdrawn,  will  reproduce  the  same  amount 
of  external  work  in  an  opposite  direction  to  that  in  which  the 
force  acted. 

NOTE  (&). 

The  work  of  a  pressure  of  constant  intensity  acting  in  the 
same  direction  as  the  path  described  by  its  point  of  applica- 
tion may  be  represented  by  a  geometrical  diagram  in  the 


EDITORIAL   APPENDIX. 


583 


same  way  as  that  used  for  representing  the  space  described 
by  a  body  moving  with  a  uniform  velocity  in  any  given 
time  ;  by  constructing  a  rectangle,  one  side  of  which  repre- 
sents the  number  of  units  of  force,  the  other  the  number  of 
units  of  path ;  the  number  of  units  of  area  of  the  rectangle 
will  express  the  number  of  units  of  work. 


NOTE  (I). 

The  method  given  (Art.  51)  for  estimating,  by  a  geometri- 
cal diagram,  the  work  of  a  pressure  which  varies  in  inten- 
sity at  different  points  of  the  path  described  in  its  line  of 
direction  by  its  point  of  application,  finds  its  application  and 
has  to  be  used  whenever  there  is  no  geometrical  law  of  con- 
tinuity by  which  the  pressure  can  be  expressed  in  terms  of 
the  path  ;  and,  even  when  such  a  law  obtains,  it  is  some- 
times found  to  be  a  more  convenient  method  of  obtaining  an 
approximate  value  of  the  amount  of  work  than  the  more 
rigorous  one  expressed  by  the  formula 

s 


S, 

in  which  U  can  be  rigorously  found  whenever  P,  which  being 
a  function  of  S  can  be  expressed  algebraically  in  terms 
of  it. 

As  an  example  of  these  two  methods  of  estimating  the 
work  of  a  variable  pressure,  acting^  in  the 
direction  of  the  rectilinear  path  described  by 
its  point  of  application,  let  the  familiar  case 
of  the  action  of  steam  on  the  piston  of  the 
steam-engine  be  taken. 

Let  ABCD  represent  the  steam-tight  cy- 
linder in  which  the  piston  is  driven  from  the 
position  at  #,  at  one  end,  to  c  at  the  other,  in 
the  direction  of  the  axis  ac,  of  the  cylinder, 
by  means  of  the  pressure  of  the  steam  on  the 
end  of  the  piston.  Let  us  suppose  that  the 
A~~  ~~B  steam  acts  with  a  constant  pressure,  repre- 

sented by  Pj,  whilst  the  piston  is  driven  through  the  portion 
ba  of  the  path,  and,  having  reached  this  point,  the  commu- 
nication between  the  cylinder  and  the  boiler  being  then  cut 


584:  EDITOEIAL   APPENDIX. 

off,  that  the  steam  already  admitted  acts,  through  the  re- 
mainder of  the  path  described  by  the  piston,  by  what  is 
termed  its  expansive  force,  in  which  the  pressure  continually 
decreases  as  the  piston  approaches  the  point  c.  Let  us  sup- 
pose that  the  law  of  variation  of  this  pressure  on  the  piston 
at  different  points  is  such  that  the  pressures  at  any  two 
points  are  inversely  proportional  to  their  distances  from  the 
point  a.  P,  then  denoting  the  pressure  when  the  piston  is 
at  &,  let  P  denote  the  pressure  when  it  has  reached  another 
point  o  at  a  distance  S  from  #,  and  S3  and  Sx  denote  the 
lengths  ac  and  ab,  then  according  to  the  above  law  there 
obtains 

P>  :  P  ::  S  :  S1?     therefore  P=PX^. 

Let  the  elementary  portion  of  the  path  be  denoted  by  <#S, 
then  by  multiplying  the  variable  force  by  the  elementary 
path  there  obtains 

P^S=Pa|^S, 

b 

which  may  be  termed  the  elementary  work,  or  in  other 
words,  the  work  done  whilst  the  variable  pressure  acts 
through  the  elementary  path,  during  which  period  the  vari- 
able pressure  may  be  regarded  as  constant. 

To  obtain  the  total  work  whilst  the  variable  pressure  acts 
from  I  to  Cj  or  through  the  path  S2—  S1?  there  obtains 

S,  8, 


.8  s,-iog.£  s,). 

s,  s,  k 

If  instead  of  the  exact  work  due  to  the  expansive  force  of 
the  steam,  and  which  is  given  by  the  foregoing  formula,  an 
approximate  value  only  was  required,  it  could  be  obtained  by 
a  geometrical  diagram  as  follows. 
Having  set  off  to  any  scale  a  num- 
ber of  units  representing  the  path 
fo,  calculate  the  pressures  at  the 
points  £,  <?,  and  at  the  middle  point 
0,  for  a  first  approximation.  That 
at  l>  will  be  simply  P1  ;  that  at  c, 

P,,  and  that  at  o,  P,         S>         ' 

2—  S,) 


TO. 

X 


T-- 

or 


EDITOKIAL   APPENDIX.  585 

Having  drawn  perpendiculars  to  ~bc  at  &,  <?,  and  <?,  set  off  on 
them  the  distances  £m,  on  and  cp  respectively  equal  to  the 
corresponding  pressures,  estimated  in  terms  of  the  unit  of 
pressure,  and  according  to  any  given  scale.  Join  mn  and 
np  ;  the  number  of  units  of  area,  in  the  figure  thus  formed, 
estimated  in  units  of  path  and  pressure,  will  be  an  approxi- 
mate value  of  the  required  number  of  units  of  work. 

The  greater  the  number  of  parts  into  which  T>c  is  divided  and 
the  corresponding  pressures  calculated,  the  nearer  will  the 
enclosed  area  approach  to  the  true  value  of  the  work. 

The  mean  pressure,  or. that  force  which,  acting  with  a 
constant  intensity  along  the  same  path  as  that  described  by 
the  point  of  application  of  the  variable  pressure,  would  give 
the  same  work,  is  found  either  by  dividing  the  result  of  the 
integration  by  83—8^  or  by  dividing  the  area  in  tlie  last 
method  by  ~bc. 

NOTE  (m). 

As  an  example  of  the  manner  of  obtaining  the  work  done 
^.  A..  by  a  constant  pressure  acting  always 

/'''         ^\  in  parallel  directions  whilst  its  inclina- 

p ^         tion  to  the  path  described  by  its  point 

of  application  is  continually  varying, 
let  the  well  known  mechanism  of  the 
crank  arm  and  connecting  rod  be  taken. 
Let  O  be  the  centre  around  which  the 
crank  arm  is  made  to  revolve,  by  the 
application  of  a  constant  pressure  Px, 
transmitted  through  a  connecting  rod 

CD,  all  of  whose  positions  during  the  motion  are  parallel  to 
the  diameter  AB.  The  path  described  by  the  point  of  ap- 
plication C  will  be  the  circumference  of  which  OC  is  the 
radius,  and  the  inclination  of  Pa  to  this  path  will  be  the 
variable  angle  DCN,  between  its  direction  and  the  tangent 
to  the  circle  at  C,  of  which  the  variable  angle  AOC,  that 
measures  the  inclination  of  the  crank  arm  to  the  diameter 
AB,  is  the  complement.  Denote  this  last  angle  by  a,  and 
the  length  of  the  crank  arm  OC  by  ~b.  Now  decomposing 
P!  into  components  in  the  direction  of  the  tangent  CN  and 
the  radius  OC,  we  obtain  for  the  first  Pj  sin.  a,  and  for  the 
second  P1  cos.  a,  of  which  Pa  sin.  a  is  alone  effective  to  pro- 
duce work,  since  Pj  cos.  a  acts  constantly  towards  the  fixed 
point  O  without  describing  any  path  in  the  direction  of  its 


f/ 


586  EDITORIAL   APPENDIX. 

action.  But  the  elementary  path  described  by  the  point 
of  application  is  evidently  Ida,  the  infinitely  small  arc  Cn  of 
the  circle.  The  elementary  work  of  the  variable  component 
P,  sin.  a  will  therefore  be  expressed  by 

Pj  sin.  a  x  Ma. 

The  total  work  for  any  portion  of  the  path,  as  AC,  will 
therefore  be 
.a 

\  sin.  a  bda=T>J>(l—co8.  a)=PJ)  ver.  sin.  a. 
o 

and  for  O=TT,  it  becomes 

Pax25,  orP,xAB; 

a  result  which  might  have  been  foreseen,  since  AB  is  the 
path  described  by  the  point  of  application  of  P,  in  its  line 
of  direction,  whilst  the  actual  path  is  the  semi-circumfe- 
rence ACB. 

As  Cn=bda,  if  through  n  a  perpendicular  nm  is  drawn  to 
CD,  the  line  of  direction  of  P1}  the  distance  Cm  is  evidently 
the  projection  of  the  elementary  path  actually  described  on 
the  line  of  direction  of  P,,  and  is  therefore  the  corresponding 
elementary  path  of  Px  in  its  line  of  direction ;  but  Cm=C?» 
sin.  a=bda  sin.  a.  Denoting  AB  by  A,  then  Cm=dh ;  and 
there  obtains 

dh=bda  sin.  TT  ;  and  P1  dh=P1  Ida  sin.  a ; 
and 


r 

J 


A  result  the  same  as  is  shown  to  obtain  by  the  preceding 
proposition. 

To  find  the  mean,  or  constant  pressure  which,  acting  in 
the  direction  of  the  circular  path,  would  produce  the  same 
amount  of  work  as  the  variable  force  does  in  acting  through 
the  semi-circumference  ;  call  Q  this  mean  force,  its  path 
being  TT&,  its  work  will  be  Q  x  tib  ;  and  as  this  is  to  be  equal 
to  the  work  of  Pt  sin.  a,  there  obtains 

Q  x  n^=Pl  x  2ft,   hence  Q^P,  -  =0*6366  P,  nearly, 
for  the  value  of  the  force. 


EDITORIAL   APPENDIX.  587 

It  may  be  well  to  observe  here  that  the  mean  pressures 
have  no  farther  relations  to  the  actual  pressures  than  as 
numerical  results  which  are  frequently  used  instead  of  the 
actual  pressures  to  facilitate  calculations;  and  also  as  a 
means  of  comparing  results,  or  work  actually  obtained  from 
a  force  of  variable  intensity,  at  different  epochs  of  its  action, 
with  what  would  have  been  yielded  at  the  same  epochs  by 
the  equivalent  mean  force. 

To  show  the  manner  of  making  the  comparison  in  this 
case,  let  us  take  the  two  expressions  for  the  quantity  of  work 
due  the  mean  force,  and  also  to  the  variable  component,  for 
a  portion  of  the  path  corresponding  to  any  angle  a.  Since 

2 
Q^Pj-,  its  work  corresponding  to  a  will  be 


The  corresponding  work  of  the  variable  component  Yl  sin.  a 
will  be 

P^CL-cos.  a). 

The  difference  therefore  between  these  two  amounts  of  work 
will  be 


.  a). 


Now  this  difference  will  be  0  for  the  following  values  of  o, 

a=0,        a=  -,       and  a=7r. 

2 

The  maximum  value  of  this  difference  can  be  found  by  the 
usual  method  of  differentiation  and  placing  the  first  differ- 
ential coefficient  equal  to  0.  Performing  this  operation, 
there  obtains 

sin.  a  =-=  0-6366; 

<r 

the  corresponding  values  of  a  being  respectively 
a  —  0-21964  ?r,and  a  =  TT  —  0-21964  n. 

Substituting  these  values  of  a  and  the  corresponding  values 
of  cos.  a  in  the  preceding  expression  for  the  difference  there 
obtains,  for  the  first, 


588  EDITORIAL   APPENDIX. 

PJ,  (^_l  +  cos.  a)  =?$  (2x0-21964-1+ 

+  0-21039^6; 
and  for  the  second, 

PJ>  (^-1-f-cos.  a)=PJ>  (2x0-21964-1-1/1-1)  = 
-  0-21039  PA 

From  these  two  expressions  it  is  seen  that  the  greatest  excess 
of  the  work  of  the  mean  force  over  that  of  the  other  would 
be  +  0-21039  P1&=+  0-1052  xP,2J,  or  about  TV  of  the  total 
work  of  Pj  corresponding  to  the  path  25  ;  whilst  that  of  the 
work  of  P,  over  the  mean  force,  represented  by  —  0'21039P1£, 
is  the  same  in  amount. 

If  now  we  suppose  the  direction  of  the  constant  force  Pa 
to  be  changed,  when  its  point  of  application  reaches  the  point 
B,  so  as  to  act  parallel  to  the  direction  BA  until  the  point  of 
application  arrives  at  A,  it  is  clear  that  the  work  rof  P^  due 
to  the  path  described  from  B  to  A  will  also  be  expressed  by 
Pt  x  25,  so  that  the  work  due  to  an  entire  revolution  of  the 
point  of  application  will  be  Pt  x  45.  As  the  mean  force  will 
evidently  be  the  same  for  the  entire  revolution  of  the  point 
of  application,  it  follows  that  the  greatest  positive,  or  nega- 
tive excess,  as  stated  above,  will  be  0-0526  xPj4J,  or  210  of 
the  work  for  one  entire  revolution. 

It  is  thus  seen  that  although  the  work  of  the  effective 
variable  component  P,  sin.  a  of  rl  is  not,  like  that  of  the  mean 
force,  uniform  for  equal  paths,  still  it  at  no  time  falls  short 
of  nor  exceeds  the  work  of  the  mean  force  by  more  than 
about  ¥V  of  the  entire  work  for  each  revolution.  Were  any 
mechanism,  as  that  for  pumping  water  for  example,  so 
arranged  that  either  the  constant  force  P15  or  a  mean  force 
equal  to  0*6366  P15  acting  as  above  described,  were  applied 
to  it,  the  quantity  of  water  delivered  by  the  one  would  at  no 
time  exceed,  in  any  one  revolution,  that  delivered  by  the 
other  by  more  than  ¥V  of  the  total  quantity  delivered  by 
either  during  the  entire  revolution. 


NOTE  (n). 

If  P2,  for  example,  were  the  resultant  of  the  other  pres- 
sures, its  component  P3  cos.  aa  would  be  equal  to  the  alge- 


EDITORIAL    APPENDIX.  589 

braic  sum  of  the  components  P!  cos.  al5  P3  cos.  «3,  &c.,  of  the 
other  pressures  P15  P3,  &c. ;  the  work  therefore  of  Pa,  esti- 
mated in  the  direction  of  the  given  path  AB,  and  corres- 
ponding to  any  portion  of  this  path,  will  be  equal  to  the 
algebraic  sum  of  the  work  of  the  other  pressures  PM  P3,  &c., 
corresponding  to  the  same  portion  of  the  given  path. 


NOTE  (0). 

Since  at  the  point  E,  taken  as  the  point  of  application,  the 
line  of  direction  of  the  pressure  becomes  a  tangent  to  the 
arc  described  with  the  radius  OE,  it  follows  that  the  infi- 
nitely small  arc  described  with  the  radius  OE  may  be  taken 
for  the  infinitely  small  path  described  by  the  point  of  appli- 
cation in  the  direction  of  the  tangent.  Denoting  by  da  the 
infinitely  small  angle  described  by  the  radius  OE,  then 
OE  x  da  will  express  the  infinitely  small  path,  or  arc  ;  and 
P  x  OE<^a  will  represent  the  elementary  work  of  the 
pressure.  ' 

If  the  pressure  remains  constant  in  intensity  and  direction 
during  an  entire  revolution  of  the  body  about  0,  then  will 
the  work  of  P  for  this  revolution  be  represented  by 
P  x  circum.  OE. 


(p). 

The  term  living  force  is  more  generally  used  with  us  by 
writers  on  mechanics  instead  of  its  Latin  equivalent  vis  vwa, 
to  designate  the  numerical  result  arising  from  multiplying 
the  quantity  denominated  the  mass  of  a  body  by  the  square 
of  the  velocity  with  which  the  body  is  moving  at  any 
instant.  It  will  be  readily  seen  that  this  product  does  not 
represent  a  pressure,  or  force,  but  the  numerical  equivalent 
of  the  product  of  a  certain  number  of  units  of  pressure  and  a 
certain  number  of  units  of  path.  The  one  magnitude  being 
of  as  totally  a  distinct  order  from  the  other  as  an  area  is 
different  from  a  line,  and  therefore  having  no  common  unit 
of  measure. 

Besides  this  expression,  which  serves  no  other  really  use- 
ful purposes  than  as  a  name  to  designate  a  certain  numerical 
magnitude  which  is  of  constant  occurrence  in  the  subject  of 
mechanics,  there  is  another  also  of  frequent  use,  termed 


590  EDITORIAL    APPENDIX. 

quantity  of  motion,  wnich  is  the  product  of  the  mass  and 


w 


the  velocity,  or — v.     This  is  also  termed  the   dynamical 

measure  of  a  force  in  contradistinction  to  pressure,  as  usually 
estimated,  which  is  termed  the  statical  measure  of  a  force. 


In  estimating  the  accumulated  work  in  the  pieces  of  a 
machine  which  have  either  a  continuous  or  a  reciprocating 
motion  of  rotation  it  is  necessary  to  find  expressions  for  the 
moments  of  inertia  of  these  pieces  with  respect  to  their  axis 
of  rotation,  and  this  may,  in  all  cases,  be  done,  within  a  cer- 
tain degree  of  approximation  to  the  true  valu£,  by  calculat- 
ing separately  the  moment  of  inertia  of  each  of  the  compo- 
nent parts  of  each  piece  and  taking  their  sum  for  its  total 
moment  of  inertia,  on  the  principle  that  these  moments  may 
be  added  to  or  subtracted  from  each  other  in  a  manner 
similar  to  that  in  which  volumes,  or  areas  are  found  from 
their  component  parts. 

In  making  these  approximate  calculations,  which  in  many 
cases  are  intricate  and  tedious,  it  will  be  well  to  keep  in  view 
the  two  or  three  leading  points  following,  with  the  examples 
given  in  illustration  ol  some  of  the  more  usual  forms  of 
rotating  pieces. 

1st.  The  general  form  for  the  moment  of  inertia  of  a  body 
rotating  around  an  axis  parallel  to  the  one  passing  through 
its  centre  of  gravity  as  given  in  equation  58,  (Art.  79)  is 


Now  if  the  distances  of  the  extreme  elements  of  the  body 
from  the  axis  passing  through  its  centre  of  gravity  are  small 
compared  with  that  of  A,  the  distance  between  the  two  axes, 
the  second  term  I  of  the  second  member  of  this  equation 
may  be  neglected  with  respect  to  the  first,  and  A2M  be  taken 
as  the  approximate  value  of  the  required  moment.  This 
consideration  will  find  its  application  in  many  of  the  cases 
referred  to,  as,  for  example,  in  that  of  finding  the  moment  of 
inertia  of  the  portion  of  a  solid,  like  the  exterior  flanch  of  the 
beam  of  a  steam-engine,  the  volume  of  which  may  be  approx- 
imately obtained  by  the  method  of  Guldinus  (Art.  39.).  In 
this  case,  A  representing  the  area  of  the  cross  section  of  the 


EDITORIAL    APPENDIX.  591 

flanch,  and  s  the  path  which  its  centre  of  gravity  would 
describe  in  moving  parallel  to  itself  in  the  direction  of  the 
flanch  around  the  beam,  any  elementary  volume  of  the 
flanch  between  two  parallel  planes  of  section  will  be  ex- 
pressed by  Ads.  Now  the  moment  of  inertia  of  this  elemen- 
tary volume  from  equation  58  is 


in  which  the  first  term  of  the  second  member,  which 
expresses  the  sum  of  the  elementary  volumes  Ads  into  the 
squares  of  their  respective  distances  r  from  the  axis  of  rota- 
tion, may  be  taken  as  the  approximate  value  required  ;  inas- 
much as  I,  the  sum  of  their  moments  of  inertia  with  respect 
to  the  parallel  axes  through  their  centres  of  gravity,  may  be 
neglected  witn  respect  to  the  first  term.  The  problem  will 
therefore  reduce  to  finding  the  moment  of  inertia  of  the  line 
represented  by  s,  which  would  be  described  by  the  centre  of 
gravity  of  A,  with  respect  to  the  assumed  axis  of  rotation, 
and  then  multiplying  the  result  by  A. 

2nd.  As  the  line  s  is  generally  contained  in  a  plane  per- 
pendicular to  the  axis  of  rotation,  and  is  given  in  kind,  as 
well  as  in  position  with  respect  to  this  axis,  being  also  gene- 
rally symmetrically  placed  with  respect  to  it,  its  required 
moment  of  inertia  may,  in  most  cases,  be  most  readily 
obtained  by  finding  the  moment  of  inertia  of  s  separately, 
with  respect  to  two  rectangular  axes  contained  in  its  plane, 
and  taken  through  the  point  in  which  the  given  axis  of  rotation 
pierces  this  plane,  and  then  adding  these  two  moments. 

The  moment  of  inertia  of  a  line  taken  in  this  way  with 
respect  to  a  point  in  its  plane  has  been  called  by  some 
writers  the  polar  moment  of  inertia. 

This  method  is  also  equally  applicable  to  finding  the  mo- 
ment of  inertia  of  a  plane  thin  disk  revolving  around  an  axis 
perpendicular  to  its  plane,  and  to  solids  which  can  be  divided 
into  equal  laminae  by  planes  passed  perpendicular  to  the  axis 
of  rotation. 


(a1)  The  moment  of  inertia  of  the  arc  of  a  parabola  with 
respect  to  an  axis  perpendicular  to  the  plane  of  the  curve  at 
a  given  point  on  the  axis  of  the  curve. 

Let  BAG  be  the  given  arc  ;  A  the  vertex  of  the  parabola  • 


592 


EDITORIAL   APPENDIX. 


R  the  point  on  its  axis  at  which  the 
axis  is  taken.  Through  R  draw 
the  chord  PQ.  Represent  the 
D  chord  BC  of  the  given  arc  by  5; 
its  corresponding  abscissa  AD  by  a\ 
and  AR  by  c.  Let  y  represent  the 
ordinate  pq,  and  x  the  correspond- 

ing abscissa  of  any  element  dz  of  the  arc. 

From  the  preceding  remarks,  the  moment  of  inertia  of  dz 

with  respect  to  the  axis  AD  will  be  expressed  by  y*dz  ;  and 

that  of  the  entire  arc  BAG  by 


as  from  the  equation  of  the  parabola,  y*=  ^  x. 
By  integration  f 


in  which  Z  is  the  length  of  the  arc  BAG. 

In  like  manner  the  moment  of  inertia  of  dz  with  respect 
to  the  chord  PQ  is 


and  for  the  entire  arc  BAG, 


i& 


*z» 


=  2      ^-«f  A  =    .- 

0  0 

which  integrated  as  above, 


6l?^9/  ^Z  U 


*  Church's  Int.  Cal.  Art.  199. 


f  Ibid.  Art.  150. 


EDITORIAL   APPENDIX.  593 

From  the  preceding  remarks,  the  moment  of  inertia  of  Z, 
with  respect  to  the  axis  at  the  point  E  perpendicular  to  the 
plane  of  Z,  is 


32 


\ 

/  " 


The  value  of  Z  in  the  above  expressions  is 


Each  of  the  preceding  expressions  may  be  simplified,  and 
an  approximate  value  obtained,  sufficiently  near  for  practical 
applications,  when  the  ratios  of  I  and  c  to  a  are  given*  For 
example,  when  b  /_  \a  there  obtains 


the  terms   omitted,  being  small  fractions  with  respect  to. 
unity,  do  not  materially  affect  the  result. 

Having  found  the  moment  of  inertia  of  a  parabolic  curve,  . 
that  of  a  parabolic  ring  of  uniform  cross  section,  taken  per- 
pendicular to  the  direction  of  the  curve  at  any  point,  and 
having  its  centre  of  gravity  at  its  point  of  intersection  with 
the  curve,  can  be  obtained  by  simply  multiplying  Ij+I,  by 
S  the  area  of  the  given  cross  section. 


(A1)  The  moment  of  inertia  of  the  segment  of  a  parabola  with 
respect  to  an  axis  perpendicular  to  its  plane  at  a  given 
point  of  the  axis  of  the  curve. 

Let  BAG  be  the  given  segment ;  A  the  vertex  ;  AD  the 

*  Church's  Int.  Cal.  Art.  199. 
38 


EDITORIAL    APPENDIX. 


axis  of  the  curve ;  and  D  the  point  on 
the  axis  with  respect  to  which  the  mo- 
ments of  inertia  are  estimated.  Denote 
the  chord  BC  by  5;  the  abscissa  AD 
by  a. 

By  (Art.  81)  the  moment  of  inertia  of 
an  elementary  area  pq  with  respect  to  AD  is  T^  (pq)*dx= 
'    That  of  the  segment  therefore  will  be 


~ 


In  like  manner  the  moment  of  inertia  of  an  elementary 
area  asps,  with  respect  to  the  axis  BC,  is  -J  (ps)3dy=  -J(a—  a?)8 
dy.  That  of  the  segment  therefore  will  be 

16  *& 


=  f  f 


(a  - 


From  this  last  expression  we  readily  obtain  the  moment 
of  inertia  of  a  disk  having  the  segment  for  its  base  and  its 
thickness  represented  by  c,  with  respect  to  an  axis  at  D  per- 
pendicular to  its  base  by  simply  multiplying  Ix  +  12  by  c  ; 

ft  +  1.)  c  =  A  «fo  +  T%  0'fo  =  I  ^ 
in  which  f  «Z»c  =  Y,  the  volume  of  the  disk. 


or 


l)  The  moment  of  inertia  of  a  parabolic  disk,  or  prism,, 
with  respect  to  an  axis  parallel  to  the  chords  which  termi- 
nate the  upper  and  lower  bases  and  midway  between  the 
chords. 

Let  pq  be  an  elementary  volume  of  the  disk  contained 
between  two  planes  parallel  to  the  base 
BC  of  the  disk.  Adopting  the  same 
notation  as  in  the  preceding  article,  the 
volume  pq  is  expressed  by 

2t/ .  c  .  dx. 

The  moment  of  inertia  of  this  elemen- 
tary volume  with  respect  to  an  axis 
through  its  centre  of  gravity  and  parallel 
to  BC  is  (Art,  83) 


EDITORIAL    APPENDIX. 

W  .  2v  .  c  .  dx 


595 


and  the  moment  of  inertia  of  the  same  volume  with  respect 
to  the  axis,  parallel  to  the  one  through  its  centre  of  gravity, 
taken  on  the  base  BC  of  the  disk  and  midway  between  the 
upper  and  lower  chords  is  (Art.  79  Eq.  58) 


.  c  .  dx 


.  c  .  dx  (a—xf  ; 


the  moment  of  inertia  of  the  entire  disk  with  respect  to  the 
same  axis  is 

.-.  I  =  Tv  C  2y  .  G  .  dx  {c*  +  (dx)*\  +  f  Zy  .  c  .  dx  (a—x)\ 


Substituting  for  x  and  dx  in  terms  of  y,  omitting  the  term 
containing  (dx)3,  and  integrating  as  indicated,  there  obtains, 


a 


in  which  V= 


(dr)  The  moments  of  inertia  of  a  right  prism  with  a  trape* 
zoidal  "base  with  respect  to  axes  perpendicular  and  parallel 
to  the  base  at  the  middle  point  oj  the  face  terminated  by  the 
broader  side  of  the  trapezoid. 

Let  AG-HC  be  the  trapezoid  forming  the  base  of  the  prism. 
Represent  the  altitude  EF  of  the  trape- 
zoid by  a ;  AG  by  I ;  CH  by  b1 ;  and  the 
height  GB  of  the  prism  by  c.  Let ^  be  an 
elementary  volume  of  the  prism  between 
two  planes  parallel  to  the  face  AB  and 
at  a  distance  Ee=x  from  the  face  CD. 
From  C  drawing  Cc  parallel  to  HG  there 
obtains 


A  /I 

pr=  ==-  .  Ac  =  -  (b 


EF 


a 


=pr 


V. 


596  EDITORIAL   APPENDIX. 

The  elementary  volume  pq  is  therefore 


The  moment  of  inertia  of  py  with  respect  to  an  axis  through 
its  centre  of  gravity  and  perpendicular  to  the  base  of  the 
prism  is  (Art.  83). 


and  that  of  the  entire  prism  with  respect  to  an  axis  at  F,  the 
middle  point  of  AGr,  and  parallel  to  the  preceding  axis,  is 


=  A  f  |  -  Q-V)+V  [  c. 

«/         I  # 


omitting  the  term  containing  (dx}\  and  integrating,  as  indi- 
cated, there  obtains 


in  which  F= 


By  a  like  series  of  operations  the  moment  of  inertia  of  the 
entire  prism,  with  respect  to  an  axis  perpendicular  to  the 
preceding  one  at  its  middle  point  between  the  upper  and 
lower  bases  of  the  prism,  will  be 


EDITORIAL    APPENDIX. 


597 


(e)  The  moment  of  inertia  of  a  right  prismoid  with  rectan- 
gular bases  with  respect  to  an  axis  XY  through  the  centre 
o£  gravity  of  the  tower  base  and  parallel  to  one  of  its 
sides. 

Let  AB  =  J,  BC  =  c  be  the  sides  of  the  rectangle  of  the 
lower  base ;  ab  =  b\  be  =  cl  the  sides 
of  the  upper  base.  Let  pqrs  be  any 
section  of  the  prismoid  parallel  to 
the  lower  base  and  at  a  distance  x 
from  it*  and  let  a  be  the  altitude  of 
the  prismoid,  or  the  distance  between 
its  upper  and  lower  bases. 

From  the  relations  between  the 
dimensions  of  the  prismoid  there  ob- 
tains (Art.  «f ') 


a 

x.                      x(c  —  cl)  +  acl 
2r=-(c-cl)+cl=-     —^ ; 

and  to  express  the  elementary  solid  contained  between  two 
planes  parallel  to  the  base  of  the  prismoid  and  at  the  height 
x  above  it, 

x(b  —  bl)  +  ab1      x(c  —  cl)  +  ac1 

— x — * - .  dx. 


The  moment  of  inertia  of  this  solid,  with  respect  to  an 
axis  xy  through  its  centre  of  gravity  and  parallel  to  XY,  is 
(Art.  83) 

x  (b  —  b1)  +  afr     x(c  —  cl}+atf 

--  -•* 


The  moment  of  inertia  of  the  prismoid  (Art.  79  Eq.  58)  is 

\  (*c-^+ 


593  EDITORIAL   APPENDIX. 


o 


fxg>- 
J  a 


omitting  the  term  containing  (d®)*  and  integrating  as  indi 
cated,  there  obtains, 


By  integrating  the  expression  for  the  elementary  volume 
between  the  same  limits,  there  obtains  to  express  the  volume 
of  the  prisinoid 


which  is  the  formula  usually  given  in  mensuration. 

In  each  of  the  preceding  examples,  the  quantities  I,  I15  &c. 
are  expressed  only  in  terms  of  certain  linear  dimensions  ;  to 
obtain  therefore  the  moments  of  inertia  proper  these  results 

must  be  multiplied  by  the  quantity  -,  or  the  unit  of  mass 

corresponding  to  the  unit  of  volume,  in  which  ^  represents 
the  weight  of  the  unit  of  volume  of  the  material  and 
g  =  32£  feet. 

Each  of  the  above  values  of  I  may  be  placed  under  more 
simple  forms  for  the  greater  readiness  of  numerical  calcula- 
tion by  throwing  out  such  terms  as  will  visibly  affect  the 
result  in  only  a  slight  degree.  But  as  such  omissions  depend 
upon  the  numerical  relations  of  the  linear  dimensions  of  the 
parts  no  rule  for  making  them  can  be  laid  down  which  will 
be  applicable  to  all  cases.  . 


(/"')  The  moment  of  inertia  of  a  trip  hammer. 

These  hammers  consist  of  a  head  of  iron  of  which  A  repre- 
sents a  side  and  A'  a 
front  elevation;  of  a 
handle  of  wood  B, 
which  is  either  of  the 
shape  of  a  rectangular 
parallelepiped,  or  of 


EDITORIAL    APPENDIX.  599 

• 

two  rectangular  prismoids,  having  a  common  base  at  the 
axis  of  rotation  C  where  the  trunnions,  upon  which  the 
hammer  revolves,  are  connected  firmly  with  the  handle  by 
an  iron  collar.  Another  iron  collar  is  placed  at  the  end  of 
the  handle,  and  is  acted  on  by  that  piece  of  the  mechanism 
which  causes  the  hammer  to  rotate. 

To  obtain  the  moment  of  inertia  of  the  whole,  that  of  each 
part  with  respect  to  the  axis  is  separately  estimated  and 
the  sum  then  taken. 

The  head  A,  A'  may  be  regarded  as  a  parallelopiped  of 
which  the  side  A',  reduced  to  its  equivalent  rectangle  by 
drawing  two  lines  parallel  to  the  vertical  line  that  bisects 
the  figure,  is  the  end,  and  the  breadth  of  the  side  A  is  the 
length.  If  then  from  the  moment  of  inertia  of  this  parallel- 
opiped that  of  the  void  a,  or  eye  of  the  -hammer,  which  is 
also  a  parallelopiped,  be  taken,  the  difference  will  be  the 
moment  of  inertia  of  the  solid  portion  of  the  head.  The 
moments  of  inertia  of  these  parallelepipeds  may  be  calcu- 
lated, with  respect  to  the  axis  C,  by  first  estimating  them 
with  respect  to  the  axes  through  their  respective  centres  of 
gravities  G  and  ^,  parallel  to  C,  by  (Art.  83)  and  then  witli 
respect  to  C  by  (Art.  79.  Eq.  58).  Or  if  the  moments  of 
inertia  with  respect  to  G  and  g  are  small  with  respect  to  the 
product  of  their  volumes  and  the  squares  of  the  distances 
GO  and  ^C,  then  the  difference  of  the  latter  products  may 
be  taken  as  the  approximate  value. 

The  moment  of  inertia  of  the  handle,  if  also  a  parallelo- 
piped, will  be  found  with  respect  to  C  by  (Arts.  79,  83).  If 
it  is  composed  of  two  rectangular  prismoids,  then  the  mo- 
ment of  the  parts  on  each  side  of  the  axis  must  be  found  by 
(«')  and  their  sum  taken. 

The  moment  of  inertia  of  the  trunnions  and  the  iron  hoop 
to  which  they  are  attached  may  be  found  by  (Arts.  85,  87) 
and  their  sum  taken.  But  as  this  quantity  will  be  generally 
small  with  respect  to  the  others  it  may  be  omitted. 

That  of  the  hoop  at  the  end  of  the  handle  may  be  taken 
approximately  as  equal  to  the  product  of  its  volume  and  the 
square  of  the  distance  between  the  axis  through  its  centre 
of  gravity  and  that  of  rotation. 


(gf)  The  moment  of  inertia  of  a  cast  iron  wheel. 
These  wheels  usually  consist  of  an  exterior  rim  A  A'  of 


600 


EDITORIAL   APPENDIX. 


uniform  cross  section  con- 
nected with  the  boss,  or  nave 
C,  C',  which  is  a  hollow 
cylinder,  by  radial  pieces,  or 
arms  B,  B',  the  cross  section 
of  which  is  in  the  form  of  a 
cross.  Each  arm  having  the 
same  breadths  at  top  and 
bottom  in  the  direction  of 
the  axis  of  the  wheel  as  those  of  the  rim  and  nave  which  it 
connects;  the  thickness  perpendicular  to  the  axis  being 
uniform.  The  projection  or  ribs  on  the  side  of  each  arm, 
and  which  give  the  cross  form  to  the  section,  being  of  uni- 
form breadth  and  thickness ;  or  else  of  uniform  thickness 
but  tapering  in  -breadth  from  the  nave  to  the  rim.  These 
ribs  join  another  of  the  same  thickness  that  projects  from  the 
inner  surface  of  the  rim. 

Eepresent  by  E  the  mean  radius  of  the  rim,  estimated  from 
the  axis  to  the  centre  of  gravity  of  its  cross  section ;  J  its 
breadth,  and  d  its  mean  thickness  ;  V  its  volume,  and  I  its 
moment  of  inertia  with  respect  to  the  axis;  ^  the  weight  of 
its  unit  of  volume,  and  y=3%%  feet ;  then  by  (Art.  86) 

V=totW>d    andl=^  FE2, 
V 

omitting  J^2  as  but  a  small  fractional  part  of  E2. 

Eepresenting  by  bl  the  breadth  of  the  arm  at  the  axis, 
supposing  it  prolonged  to  this  line  ;  J2  its  breadth  at  the  rim, 
supposing  it  prolonged  also  to  the  mean  circle  of  the  rim,  d1 
its  thickness  ;  V.  its  volume  ;  I,  its  moment  of  inertia,  then 


2 


Eepresenting  by  a^  the  breadth  at  bottom,  «2  the  breadth  at 
top  of  the  ribs,  or  projections  on  the  sides  of  each  arm,  esti- 
mated also  at  the  axis  and  mean  circle  of  the  rim  ;  d^  their 
thickness  ;  F2  their  volume  ;  I2  their  moment  of  inertia  ; 
then  by  (a') 

F.=B*  ?dp      and  !.=£ 


The  sum  I  +1,4-1,  will  be  the  moment  of  inertia  of  the 


EDITORIAL   APPENDIX.  601 

entire  wheel  approximately,  since  the  moment  of  inertia  of 
the  portions  of  the  boss  between  the  arms  is  omitted,  this 
being  compensated  for  by  supposing  the  arms  prolonged  to 
the  axis  and  to  the  mean  circle  of  the  rim.  As  the  quanti- 
ties Fj,  F"2,  Ij  and  I2  are  taken  but  for  one  arm,  they  must 
be  multiplied  by  the  number  of  arms  to  have  the  entire 
moment. 

(hf)  The  moment  of  inertia  of  a  cast  iron  steam  engine  beam. 

These  beams  usually  consist  of  two  equal  arms  symmetri- 
cal with  respect  to  a 
line  a  a'  through  the 
axis  of  rotation  o. 
Each  arm,  a  V  a'  and 
a  1)  a',  consists  of  a 
parabolic  disk  of  uniform  thickness  ;  b  and  V  being  the  ver- 
tices of  the  exterior  bounding  curves,  a  a'  their  common 
chord,  and  ob,  ob'  their  axes.  The  disk  is  terminated  on  the 
exterior  by  a  flanch  B  of  uniform  breadth  and  thickness.  A 
rib  C,  either  of  uniform  breadth  and  thickness,  or  else  of 
uniform  thickness,  and  tapering  in  breadth  from  the  centre  o 
to  the  ends  &,  5',  projects  from  each  face  of  the  plane  disk 
along  the  axis  b  b'.  The  beam  is  perforated  at  the  centre, 
near  the  two  extremities  and  at  intermediate  points,  to 
receive  the  short  shafts,  or  centres  around  which  rotation 
takes  place.  Around  each  of  these  perforations,  projections, 
or  bosses  D',  D",  &c.,  are'  cast,  to  add  strength  and  give  a 
more  secure  fastening  for  the  shafts. 

The  beam  being  symmetrical  with  respect  to  a  a',  it  will 
be  only  necessary  to  calculate  the  moments  of  inertia  of  the 
component  parts  of  each  arm  with  respect  to  the  axis  o  and 
take  double  their  sum  for  the  total  moment  of  inertia  of  the 
beam.  These  component  parts  are — 1st,  the  parabolic  flanch ; 
2nd,  the  parabolic  disk  of  uniform  thickness  enclosed  by  the 
flanch ;  3d,  the  rib  on  each  side  of  the  disk,  running  along 
the  central»line  bbf ;  4th,  the  projections,  or  bosses  D'  &c., 
around  the  centres. 

The  moment  of  inertia  of  the  flanch  will  be  calculated  by 
(«')  as  its  thickness  is  small  compared  with  the  other  linear 
dimensions.  That  of  the  disk  will  be  calculated  by  (&'). 
That  of  the  rib  by  (df).  Those  of  the  projections  may  be 
obtained  within  a  sufficient  degree  of  approximation  by 


602  EDITORIAL   APPENDIX. 

taking  the  product  of  their  volumes  and  the  squares  of  their 
respective  distances  from  the  axis  o. 

The  sum  of  these  quantities  being  taken  it  must  be  multi- 
plied by  -  as  in  the  preceding  cases ;  v>  being  the  weight  of 
the  unit  of  volume  of  the  material. 


NOTE  (s). 

The  increase  of  tension  due  to  rigidity  and  which  ig  ex- 
pressed by  -—  -  —  -  —  -  may  be  placed  under  the  following 

form, 

cm  .  a+cm  .  I  .  Pa        cm  (a+l  .  Fa) 

K  R 

by  writing  cm  .  a  for  D,  and  cm  .  ~b  for  E,  in  which  c  repre- 
sents the  circumference  of  the  rope,  and  m  the  power  to 
which  c  is  raised. 

The  increase  of  tension  of  any  other  rope  whose  circumfer- 
ence is  cl  bent  over  the  same  pulley  and  subjected  to  the 
same  tension  Pa  is,  in  like  manner,  expressed  by 


E 

Now  representing  by  T  and  Tl  the  two  values  above  for  the 
respective  increase  of  tension  for  c  and  cl  there  obtains,  by 
dividing  the  one  by  the  other, 


which  expresses  the  rule  given  above  for  using  the  tables  in 
calculating  the  increase  of  rigidity  due  to  a  cord  whose  cir- 
cumference is  different  from  those  in  the  tables. 


NOTE  (t). 

As  one  of  the  chief  ends  of  every  machine  designed  for 
industrial  purposes  is,  under  certain  restrictions  as  to  the 


EDITORIAL   APPENDIX.  603 

quality,  to  yield  the  greatest  amount  of  its  products  for  the 
motive  power  consumed,  it  becomes  a  subject  of  prime 
importance  to  see  clearly  in  what  way  the  work  yielded  by 
the  motive  power  to  the  receiver,  at  its  applied  point,  is 
diminished  by  the  various  prejudicial  resistances,  in  its 
transmission  through  the  material  elements  of  the  machine 
to  the  operator,  or  tool  by  which  the  products  in  question 
are  formed. 

The  most  convenient  method  for  doing  this  will  be  to 
place  (equation  112,  Art.  145)  which  expresses  the  relation 
between  the  work  2U1  of  the  motive  power  at  the  applied 
point  and  that  2U3  the  work  of  the  operator  at  the  working 

point,  with  the  portion  2U  +  -  -  2w  (v*—v*)  which  repre- 
sents the  work  consumed  by  the  prejudicial  resistances  and 
the  inertia,  under  a  form  such  that  the  work  of  each  preju- 
dicial resistance  shall  be  separately  exhibited,  for  the  pur- 
pose of  deducing,  from  this  new  form  of  the  equation,  the 
influence  which  each  of  these  has  in  diminishing  the  work 
yielded  at  the  applied  point  and  transmitted  to  the  operator. 
To  effect  this  change  of  form  in  (equation  112)  designate  by 
P!  the  motive  power,  and  Sx  the  path  passed  over  by  its 
point  of  application  in  its  line  of  direction  between  any  two 
intervals  of  time,  during  which  Pj  may  be  regarded  as  vari- 
able both  in  intensity  and  direction ;  P2  and  Sa  the  resistance 
and  corresponding  path  at  the  working  point ;  R  the  various 
prejudicial  resistances  which,  like  friction,  the  stiffness  of 
cordage,  &c.,  act  with  a  constant  intensity,  or  are  propor- 
tional to  P1?  and  S  their  path ;  wl  the  weight  of  the  parts  the 
centre  of  gravity  of  which  has  changed  its  level  during  the 

period  considered,  and  H  its  path  ;  and  —  w  (v.?—v*)=  %m 

2<7 

(v^—v^)  the  half  of  the  difference  between  the  living  forces 
or  the  accumulated  work  of  the  material  elements  in  motion, 

of  which  m  —  -  -   is  the  mass,  during  the  same  period,  in 

which  the  velocity  has  changed  from  v1  to  vv 

Now  for  an  elementary  period  dt  of  time,  during  which 
the  forces  P,  &c.,  may  be  regarded  as  constant,  and  their 
points  of  application  to  have  described  the  elementary  paths 
dSl  &c.,  in  their  lines  of  direction,  (equation  112)  will  take 
the  form, 

h,  .    .  .  (A), 


604:  EDITORIAL    APPENDIX. 

in  which  the  1st  member  of  the  equation  expresses  the  incre- 
ment of  the  living  force,  or  the  elementary  accumulated 
work  for  the  interval  dt  at  any  instant  when  the  velocity  of 
the  mass  m,  is  v\  and  the  2nd  member  the  corresponding 
algebraic  sum  of  the  elementary  work  of  Pn  R,  &c.  This 
equation  being  integrated  between  the  limits  ^  and  £2  in 
which  v  changes  from  vl  to  v^  there  obtains, 


2  lm  «-  O  = 

2  fw.dh  .....  (B). 

This   equation  (B)  is  the  same  as  (equation  112).     The 
symbol  2  designating  the   aggregate  of  the  work   of  the 

various  forces  of  the  same  kind  ;  and  that  as    /  P^Sj  &c. 

the  work  of  each  force  as  P,,  supposing  it  to  be  either  con- 
stant or  variable.  In  either  case  whenever  Px  &c.,  can  be 

expressed  in  terms  of  B!  the  value  of  /  Y1dS1  can  be  found 

by  one  of  the  methods  in  (Notes  I  and  m)  ;  and  supposing  P1 
&c.,  to  represent  their  mean  values,  and  Sx  &c.,  the  paths 
described  in  their  true  directions  during  the  interval  con- 
sidered, equation  (B)  may  be  written  under  the  following  form 
for  the  convenience  of  discussion, 


.  .  .  .(0). 

In  this  last  equation  2  /  wl  dh  =  WH  (Art.   60)  represents 

the  work  of  the  total  weight  of  the  parts  whose  centre  of 
gravity  has  changed  its  level  during  the  interval  considered, 
and  it  takes  the  double  sign  ±,  as  the  path  H  may  be 
described  either  in  the  same,  or  a  contrary  direction  to  that 
in  which  W  always  acts. 

Before  proceeding  to  discuss  the  terms  of  (equation  C), 
it  may  be  well  to  remark  that  the  term  —  RS  does  not  take 
into  account  the  work  expended  by  P,  in  overcoming  the 
molecular  forces  brought  into  play  by  the  deflection,  torsion, 
extension,  &c.,  of  the  parts  of  the  machine  ;  for,  owing  to 
the  rigidity  of  these  parts,  this  forms  but  a  very  small  .frac- 
tional part  of  the  total  work  of  the  exterior  forces  whilst  the 
machine  operates  continuously  for  some  time;  as,  during 


EDITORIAL    APPENDIX.  605 

this  time,  the  tension  of  the  parts,  or  the  molecular  resist 
ances  remain  sensibly  the  same,  and  the  molecular  displace 
ments  are  for  the  most  part  inappreciable,  or  else  very  small 
compared  with  the  paths  described  by  the  points  of  applica- 
tion of  the  other  forces. 

This  remark,  however,  does  not  apply  to  the  expenditure 
of  work  by  the  motive  power  where  the  operation  of  the 
machine  requires  that  some  of  the  parts  in  motion  shall  be 
brought  into  contact  with  others  which  are  either  at  rest,  or 
moving  with  a  slower  velocity  so  as  to  produce  a  shock. 
In  this  case  there  may  be  a  very  appreciable  amount  of 
living  force,  or  accumulated  work  destroyed  by  the  shook, 
owing  to  the  constitution  of  the  material  of  which  the  parts 
are  composed  where  the  shock  takes  place  ;  and,  if  the  shocks 
are  frequent  during  the  interval  considered,  and  in  which 
the  other  forces  continue  to  act,  the  accumulated  work 
destroyed  during  this  interval  may  form  a  large  portion  of 
the  work  expended,  or  to  be  supplied  by  the  motive  power. 
In  calculating  this  amount  of  accumulated  work  destroyed, 
we  admit  what  is  in  fact  frue  in  such  machines,  that  the 
interval  in  which  the  shock  takes  place  is  infinitely  small 
compared  with  the  interval  in  which  the  other  forces  act 
continuously,  and  therefore,  in  estimating  the  accumulated 
work  destroyed  in  each  shock,  that  we  can  leave  out  of 
account  the  work  of  the  other  forces  during  this  infinitely 
small  interval.  In  this  way,  considering  also  that  the  parts 
where  the  shock  takes  place  are  usually  formed  of  materials 
which  undergo  an  almost  inappreciable  change  of  form  from 
the  shock,  and  that  therefore  the  mechanical  combinations 
of  the  machine  are  sensibly  the  same  after  the  shock  as 
before  it,  we  readily  see  that,  to  obtain  the  total  expenditure 
of  work  by  the  motive  power,  for  any  finite  interval,  we  must 
calculate  that  which  is  consumed  by  all  the  other  resistances 
during  this  interval,  and  add  to  this  that  destroyed  by  the 
shocks  during  the  same  interval,  the  latter  being  calculated 
irrespective  of  the  work  of  the  other  forces  during  the  short 
duration  in  which  each  shock  occurs. 

We  thus  see  that,  except  in  some  cases  where  the  great 
velocity  of  the  parts  in  motion  may  give  rise  to  an  appreci- 
able expenditure  of  work  caused  by  the  resistance  of  the 
medium  in  which  these  parts  may  be  moving,  as  the  air,  &c., 
the  forces  which  act  upon  any  machine  in  motion  are  the 
motive  power ;  the  resistances,  such  as  friction,  stiffness  of 
cordage,  &c.,  which  act  either  with  a  constant  intensity 
during  the  motion,  or  are  proportional  to  the  motive  power , 


606  EDITORIAL   APPENDIX. 

the  weight  of  the  parts  whose  centres  of  gravity  do  not  remain 
on  the  same  level  during  this  interval ;  the  useful^  resistance 
arising  from  the  mechanical  functions  the  machine  is  designed 
for  ;  and  the  forces  of  inertia  which  either  give  rise  to  accu- 
mulated work,  or  the  reverse,  as  the  velocity  increases,  or 
decreases  during  the  interval  considered. 

Eesuming  equation  (C)  we  obtain,  by  transposition, 

P&=P&— I^WH-Jwv.'  +  iw^1. 

That  is  the  useful  work,  or  that  yielded  at  the  working  point 
and  which  it  is  generally  the  object  of  the  machine  to  make 
as  great  as  possible  consistently  with  the  quality  of  the 
required  products,  will  be  the  greater  as  the  terms  in  the 
second  member  of  the  equation  affected  with  the  negative 
sign  are  the  smaller. 

Taking  the  term  —  ES,  it  is  apparent  that  all  that  can  be 
done  is  to  endeavor  in  the  case  of  each  machine  to  give 
such  forms,  dimensions  and  velocities  to  those  parts  where 
these  resistances  are  developed  as  will  make  it  the  least 
possible. 

With  respect  to  WH  it  will  entirely  disappear  from  the 
equation  when  H=o  ;  in  which  case  the  centre  of  gravity  of 
the  entire  system  will  remain  at  the  same  level ;  or  else 
only  that  portion  of  this  term  will  disappear  which  belongs 
to  those  parts  of  the  machine  whose  centres  of  gravity  either 
remain  at  rest,  as  in  the  case  of  wheels  exactly  centered,  end- 
less bands  and  chains,  &c.  ;  or  in  the  case  of  those  pieces 
which  receive  a  motion  simply  in  a  horizontal  direction. 
This  term  will  also  disappear  in  whole,  or  in  part,  in  those 
cases  where  the  centre  of  gravity  ascends  and  descends 
exactly  the  same  vertical  distance  in  the  interval  correspond- 
ing to  the  work  Pfil ;  for  during  the  ascent,  as  the  direction 
of  the  path  H  is  opposite  to  that  of  the  weight  "W",  the  work 
consumed  will  be  — WH,  whereas,  in  the  descent,  it  will 
restore  the  same  amount  or  -f-WH,  and  the  sum  of  the  two 
will  therefore  be  0.  This  takes  places  in  the  parts  of  many 
machines,  for  example  in  crank  arms,  and  in  wheels  which 
are  not,  accurately  centered;  in  both  of  which  cases  the 
centre  of  gravity  ascends  and  descends  the  same  distance 
vertically  in  the  interval  corresponding  to  each  revolution 
of  these  parts  whilst  in  motion  ;  also  in  those  parts  of  a  ma- 
chine, like  the  saw  and  its  frame  in  the  saw  mill,  which  rise 
and  fall  alternately  the  same  distance. 

In  all  of  these  cases  then  the  useful  work  P2Sa  will  not  be 


EDITORIAL    APPENDIX.  607 

affected  by  the  work  due  to  the  weight  of  the  parts  in 
question. 

It  may  be  well  to  observe  that  the  preceding  remarks  refer 
only  to  the  direct  influence  of  the  weight  of  the  parts  on  the 
amount  of  useful  work ;  but  whilst  directly  it  may  produce 
no  effect  however  great  its  amount,  the  weight,  indirectly, 
may  cause  a  considerable  diminution  of  this  work,  by 
increasing  the  passive  resistances  and  thus  the  term  US. 
The  same  holds  with  regard  to  the  accumulated  work,  repre- 
sented by  the  term  %mv*,  from  which  a  considerable  dimi- 
nution may  be  made  in  P2S2  if  this  accumulated  work  cannot 
be  converted  into  useful  work,  and  thus  be  made  to  form  a 
portion  of  P2S2,  when  the  action  of  the  motive  power  is  either 
withdrawn,  or  ceases,  by  variations  in  its  intensity,  to  yield 
an  amount  of  work  which  shall  suffice  for  the  work  consumed 
by  the  resistances. 

These  last  remarks  naturally  lead  us  to  the  consideration 
of  the  two  terms  fawv*,  and  -~ jmi>22,  or  half  the  living  forces. 
or  accumulated  work  at  the  commencement  and  end  of  the 
interval  considered.  As  the  machine  necessarily  starts  from 
a  state  of  rest  under  the  action  of  the  motive  power  Pl5  it 
follows  that  fynv*,  the  accumulated  work  due  to  this  action 
tends  to  increase  P2S2,  whilst  that  —fynv*  is  so  much  accu- 
mulated in  the  moving  parts  by  which  P2S2  is  lessened. 
This  diminution  of  P2S2  is  but  inconsiderable  in  comparison 
with  the  total  useful  work  when  the  interval  in  question,  and 
during  which  the  machine  operates  without  intermission,  is 
great ;  also  in  cases  where  the  velocity  attained  by  the  parts 
in  motion  is  inconsiderable,  as  for  example  in  machines  em- 
ployed for  raising  heavy  weights,  in  which  &nv*  will  in 
most  cases  be  but  a  small  fraction  of  the  useful  work  which 
is  the  product  of  the  weight  raised  and  the  vertical  height  it 
passes  through.  In  this  last  example  we  also  see  the  incon- 
veniences which  would  result  from  allowing  bodies  raised  by 
machinery  to  acquire  any  considerable  amount  of  velocity  ; 
or  to  quit  the  machine  with  any  acquired  velocity,  as,  in 
this  case,  the  accumulated  work  generally  would  be  entirely 
lost  so  far  as  the  required  useful  effect  is  concerned. 

Except  in  the  case  where  the  accumulated  work  fynv* 
can  be  usefully  employed  in  continuing  the  motion  of  the 
machine  and  gradually  bringing  it  to  a  state  of  rest  when  the 
motive  power  P,  has  either  ceased  to  act,  or  has  so  far 
decreased  in  intensity  as  to  be  incapable  of  overcoming  the 
resistances,  whatever  tends  to  any  augmentation  of  living 
force  should  be  avoided,  for  the  teim  which  represents  this 


608  EDITORIAL   APPENDIX. 

being  composed  of  two  factors  the  one  representing  the  mass 
of  the  parts  in  motion  and  the  other  the  square  of  its  velo- 
city, it  is  evident  that  the  prejudicial  resistances  such  as 
friction  on  the  one  hand  and  the  resistance  of  the  air  on  the 
other  will  increase  as  either  of  these  factors  is  increased,  and 
thus  a  very  appreciable  amount  of  this  accumulated  work 
may  be  consumed  in  useless  work  caused  by  the  very  in- 
crease in  question.  If,  moreover,  the  machine  from  the  nature 
of  its  operations  is  one  that  requires  to  be  brought  suddenly 
to  a  state  of  rest,  any  considerable  amount  of  accumulated 
work  might  so  increase  the  effects' of  shocks  at  the  points  of 
articulation  as  to  endanger  the  safety  of  the  parts. 

The  foregoing  remarks  apply  only  to  those  parts  of  a  ma- 
chine where  the  direction  of  motion  remains  the  same  whilst 
the  machine  is  in  operation.  Where  any  of  the  parts  have 
a  reciprocating  motion,  in  which  case  whilst  the  part  is 
moving  in  one  direction  the  velocity  increases  from  0  up  to 
a  certain  limit  and  then  decreases  until  it  again  becomes  0 
at  the  moment  when  the  change  in  the  direction  of  motion 
takes  place,  and  so  on  for  each  period  of  change,  it  will  be 
readily  seen  that  where  the  velocity  varies  by  insensible 
degrees,  the  accumulated  work  of  these  parts  for  each  period 
of  change  will  be  0  and  will  therefore  have  no  influence  on 
the  amount  P2S2  of  useful  work. 

The  avoidance  of  abrupt  changes  of  velocity  in  any  of  the 
parts  of  a  machine  is  of  great  importance.  The  mechanism 
therefore  should,  as  a  general  rule,  be  so  contrived  that  there 
shall  be  the  least  play  possible  at  the  articulations  of  the 
various  parts,  and  that  the  articulations  shall  receive  such 
forms  as  to  procure  a  continuous  motion.  In  cases  also 
where  any  of  the  parts  have  a  reciprocating  motion  such 
mechanical  contrivances  should  be  used  as  will  cause  the 
variations  of  velocity  in  these  parts,  within  the  range  of 
their  paths,  to  take  place  in  a  very  gradual  manner ;  such 
for  examples  as  what  obtains  in  the  cranks  and  eccentrics 
which  are  mostly  employed  to  convert  the  continuous  circu- 
lar motion  of  one  part  into  reciprocating  motion  in  another, 
or  the  reverse. 

There  are  some  industrial  operations  however  which  are 
performed  by  shocks,  as  in  stamping  machines,  trip  ham- 
mers, &c.,  and  in  these  cases  the  useful  work  is  due  to  the 
work  developed  by  the  motive  power  in  raising  the  pestle 
of  the  stamping  machine,  or  the  head  of  the  trip  hammer 
through  a  certain  vertical  distance  from  which  it  again  falls 
upon  the  matter  to  be  acted  on,  having  acquired  in  its 


EDITORIAL    APPENDIX.  609 

descent  an  amount  of  living  force,  or  accumulated  work  due 
to  the  height  through  which  it  has  been  raised.  In  such 
cases  it  is  to  be  noted  that,  independently  of  the  work  due 
to  the  motive  power  consumed  by  the  resistances  whilst  the 
hammer  or  pestle  is  kept  in  motion  by  the  other  parts  of  the 
mechanism,  and  which  is  so  much  uselessly  consumed  so  for 
as  the  useful  work  is  concerned,  there  will  be  a  portion  of 
the  accumulated  work  in  the  pestle,  or  hammer  also  uselessly 
consumed,  arising  from  the  want  of  perfect  rigidity  and 
elasticity  in  the  material  of  which  these  two  pieces  are 
usually  composed.  Besides  this,  both  the  pestle  and  matter 
acted  on  may  and  generally  do  have  relative  velocities  after 
the  shock  between  them,  which  as  they  are  foreign  to  the 
purpose  of  the  operation,  will  also  represent  an  amount  o£* 
accumulated  work  lost  to  the  useful  work.  From  this,  w$- 
may  infer  that,  as  a  general  rule,  other  industrial  modes,  of 
operating  a  change  of  form  in  matter  will  be  preferable;  to- 
those  by  shocks,  whenever  they  can  be  employed  ;  and  that 
such  modes  are  moreover  advantageous,  as  they  avoid  those 
jars  to  the  entire  mechanism  which  accompany  abrupt 
changes  in  the  velocity  of  any  of  the  parts,  and  which,  by 
loosening  the  articulations  more  and  more,  increase  the  evil, 
and  ultimately  render  the  machine  unfit  for  service. 

Having  examined  the  influence  of  all  the  various  hurtful 
resistances  brought  into  action  in  the  motion  of  machines 
upon  the  work  PjS,  expended  by  the  motive  power,  and 
pointed  out  generally  how  the  consumption  of  the  work  may 
be  lessened,  and  the  useful  work  to  the  same  extent  increased, 
we  readily  infer  that  like  observations  are  applicable  to  the 
term  P2S2  the  work  of  the  resistance  at  the  working  point. 
As  the  prime  object  in  all  industrial  operations  performed  by 
machinery  is  to  produce  the  greatest  result  of  a  certain  kind 
for  the  amount  of  work  expended  by  the  motive  power,  it 
will  be  necessary  to  this  end  that  the  velocity,  the  form,  &c., 
of  the  operator,  or  tool  by  which  the  result  sought  is  to  be 
obtained,  should  be  such  as  will  not  cause  any  useless  expen- 
diture of  work.  On  this  point  experiment  has  shown  that 
for  certain  operators  there  is  a  certain  velocity  of  motion 
by  which  the  result  produced  will  be  the  most  advantageous 
both  as  to  the  quality  and  quantity. 

With  respect  to  the  work  of  the  motive  power  itself  repre- 
sented by  the  product  PjS,  it  admits  of  a  maximum  value ; 
for  when  the  receiver  to  which  Pj  is  applied  is  at  rest,  'Pl  will 
act  with  its  greatest  intensity,  but  the  velocity  then  being  0 
the  product  PlSl  will  also  be  0 ;  but  as  the  velocity  increases 

39 


610  EDITORIAL   APPENDIX. 

after  the  receiver  begins  to  move  the  intensity  of  the  action 
of  Px  upon  it  decreases,  until  finally  the  velocity  of  the 
applied  point  may  receive  such  a  value  Y  that  P1  will  become 
0,  and  the  product  P^  in  this  case  will  then  also  he  0.  As 
the  work  PjS,  thus  becomes  0  in  these  two  states  of  the  velo- 
city, it  is  evident  that  there  is  a  certain  value  of  the  velo- 
city which  will  make  FjS,  a  .maximum.  To  attain  this 
maximum  the  mode  of  action  of  the  motive  power  selected 
on  each  form  of  receiver  to  which  it  is  applicable  will  require 
to  be  studied,  and  such  an  arrangement  of  its  mechanism 
adopted  as  will  prevent  any  decompositions  of  the  motive 
power  that  would  tend  in  any  manner  to  increase  the  hurt- 
ful resistances  and  thus  diminish  the  useful  work. 

It  will  be  very  easy  to  show  that  the  laws  of  motion  of  all 
machines,  that  is  the  relations  between  the  times,  spaces  and 
velocities  of  the  motion  of  any  one  of  the  moving  parts  are 
implicitly  contained  in  the  general  equation  of  living  forces 
as  applied  to  machines  wThich  has  just  been  discussed. 
Resuming  (equation  B)  with  this  view,  and  representing  by 
dm  any  elementary  mass  in  motion  whose  velocity  is  v9  at 
any  instant  wrhen  it  has  described  the  path,  or  space  s,  if  we 
take  any  other  elementary  mass  dm  in  a  given  position  and 
denote  by  u  its  velocity  at  tke  same  instant,  we  shall  have 
i'a=Ma  (9,9),  and  vl=ul  ($$t)  ;  in  which  95  is  a  purely  geome- 
trical function,  since,  from  the  connection  of  the  parts  of  a 
machine,  in  which,  any  motion  given  to  one  part  is  trans- 
mitted in  an  invariable  manner  to  the  other,  the  space  passed 
over  by  any  one  point  can  always  be  expressed  in  terms  of 
that  passed  over  by  any  other  assumed  at  pleasure. 

From  the  relations  vt=ut  (95),  and  v^  dt—ds,  we  obtain 


—  v*     and  u^  d^  ^ 

(tt 

Substituting  these  values  of  v?  and  i>2  dvt  in  (equations  B  and 
A),  and  letting  m  still  represent  the  sum  of  the  elementary 
masses  as  dm,  there  obtain  the  two  equations 


tf^  /^P^S,  —2  fudS  - 
f  P2d  S2  ±  zfwdh.     (B') 


EDITORIAL    APPENDIX.  611 

ds  =  sP^-sRdS  - 
.     (A'), 

the  first  showing  the  relations  between  any  two  states  of  the 
velocities  u^  and  ut  for  any  definite  interval,  and  the  second 
for  the  infinitely  small  interval  dt.  ]STow  as  the  relations 
between  the  quantities  dS^  dS»  &c.,  or  the  elementary 
paths  described  by  the  points  of  application  of  P15  P2,  &c., 
and  the  elementary  space  ds,  from  the  connection  of  the 
parts  of  the  machine,  can  be  expressed  in  functions  of  s  and 
of  the  constants  that  determine  the  relative  magnitudes  and 
positions  of  those  parts  ;  and  as,  moreover,  P15  P2,  &c.,  are 
either  constant,  or  vary  according  to  certain  laws  by  which 
they  are  given  in  functions  of  the  paths  S1?  S2,  &c.,  we  see 
that  all  the  relations  in  question  are  implicitly  contained  in 
the  two  preceding  equations. 

Let  us  examine  the  kinds  of  motion  of  which  a  machine  is 
susceptible  and  the  conditions  attendant  upon  them.  We 
observe,  in  the  first  place,  supposing  the  machine  to  start 
from  a  state  of  rest,  that  the  elementary  work  ^ld^>l  of  the 
motive  power  must  be  greater  than  that  of  the  resistances 
combined,  or  P^Sj— R^S— &c.  >0,  so  long  as  the  velocity 
is  on  the  increase.  The  living  force  is  thus  increased  at 
each  instant  by  a  quantity  d  (mv*)=2mvdv,  or  by  an  amount 
which  is  equal  to  twice  the  elementary  work  of  the  motive 
power  and  resistances  combined ;  and  this  increase  will  go 
on  so  long  as  the  elementary  work  of  the  motive  power  is 
greater  than  that  of  the  resistances.  But,  from  the  very 
nature  of  the  question,  this  increase  cannot  go  on  indefinitely, 
for  the  point  of  application  of  the  motive  power  would  in  the 
end  acquire  a  velocity  so  great  that  P1  would  exert  no  effort 
on  the  receiver,  whereas  the  resistances  still  act  as  at  the 
commencement,  and  some  of  them  even  increase  in  intensity 
with  the  velocity.  The  living  force  therefore  will,  at  some 
period  of  the  motion,  attain  a  limit  beyond  which  it  will  not 
increase,  a  fact  which  the  operation  of  all  known  machines 
confirms,  and,  having  thus  reached  this  state,  it  must  either 
continue  the  same  during  the  remainder  of  the  time  that  the 
machine  continues  in  motion,  or  else  it  must  commence  to 
decrease  until  the  velocity  attains  some  inferior  limit  from 
which  it  will  again  commence  to  increase,  and  so  on  for  each 
successive  period  of  motion,  during  which  the  action  of  the 
forces  remains  the  same. 


612  EDITORIAL   APPENDIX. 

Supposing  the  machine  to  continue  its  motion  with  the  velo- 
city it  has  attained  at  this  maximum  state  of  the  living  force, 
we  shall  then  have 


and 


inasmuch  as  the  motion  being  now  uniform  the  difference 
between  the  living  forces  corresponding  to  any  finite  inter- 
val of  time  is  0.  Considering  the  manner  in  which  the  parts 
of  machines  are  combined  to  transmit  motion  from  point  to 
point,  we  infer  that  this  condition  with  respect  to  the 
increase  of  living  force,  and  which  constitutes  uniform 
motion,  can  only  obtain  when  the  velocities  of  all  the  differ- 
ent parts  bear  a  constant  ratio  to  each  -other.  Representing 
by  t/,  v",  «/",  &c.,  these  velocities  which  are  respectively 

equal  to  —  —  ,    _,    —  -,  &c.,  we  see  that  the  ratios  of  ds, 
dt       dt       dt 

ds",  ds'",  &c.,  will  also  be  constant  when  those  of  -y',  v",  &c., 
are  so  ;  that  is,  this  constancy  of  the  ratio  of  the  effective  velo- 
cities and  of  the  quantities  ds',  ds",  &c.,  must  subsist  together 
for  all  positions  of  the  parts  of  machines  to  which  they  refer  ; 
but  as  the  latter,  which  are  the  virtual  velocities,  or  ele- 
mentary paths  described,  depend  entirely  on  the  geometrical 
laws  that  govern  the  motion  of  the  parts,  a  little  consideration 
of  the  various  mechanical  combinations  by  which  motion  is 
transmitted  will  show  that,  in  order  that  their  ratios  shall 
respectively  remain  constant,  no  pieces  having  a  reciprocat- 
ing motion  can  enter  into  the  composition  of  the  machine, 
as  the  velocities  of  such  pieces  evidently  cannot  be  anade  to 
bear  a  constant  ratio  to  the  others.  This  condition  it  will  be 
seen  refers  exclusively  to  the  mechanism  of  the  machine,  or 
the  geometrical  conditions  by  which  the  parts  are  connected, 
and  has  nothing  to  do  with  the  action  of  the  forces  them- 
selves. 

But  when  the  condition  of  uniform  motion  is  satisfied 
there  obtains  also 


that  is,  according  to  the  principle  of  virtual  velocities,  an 
equilibrium  obtains  between  the  forces  which  act  on  the 
machine  irrespective  of  the  inertia  of  the  parts.  As  a  gene- 
ral rule  this  condition  requires  that  not  only  must  the  forces 


EDITORIAL    APPENDIX.  613 

P15  11,  &c.,  be  constant  both  in  intensity  and  direction  and 
act  continuously,  but  that  the  term  Wc$tl  must  be  sepa- 
rately equal  to  0,  or  the  centre  of  gravity  of  each  part  must 
preserve  the  same  level  during  the  motion ;  for  were  this 
not  so  any  piece  whose  weight  is  w  would  evidently  impress 
an  elementary  work  represented  by  ±wdh  which  would  be 
variable  in  the  different  positions  of  the  mechanism ;  unless 
w,  having  itself  a  uniform  velocity,  formed,  as  might  be  the 
case,  a  part  of  the  motive  power  P^  or  of  the  useful  resist- 
ance P2. 

It  thus  appears  that  to  obtain  uniform  motion  not  only 
must  the  mechanism  used  for  transmitting  the  motion  con- 
tain no  reciprocating  pieces,  and  therefore  consist  solely 
of  rotating  parts,  as  wheels,  &c.,  and  parts  moving  continu- 
ously in  the  same  direction,  as  endless  bands,  and  chains,  &c. ; 
but  that  the  centres  of  gravity  of  these  pieces  shall  remain 
at  the  same  level  during  the  motion,  which  will  require  that 
the  wheels  and  other  rotating  pieces  shall  be  accurately  cen- 
tered so  as  to  turn  truly  about  their  axes. 

The  difficulty  of  obtaining  a  strictly  uniform  motion  in 
machines  is  thus  apparent,  for  it  involves  conditions  in  them- 
selves practically  unattainable,  that  is,  applied  forces  acting 
continuously  and  with  a  constant  intensity  and  direction,  and 
that  the  ratio  of  the  virtual  velocities  of  the  different  parts 
should  be  constant  and  independent  of  the  positions  of  the 
mechanism,  a  condition  which  requires  that  the  terms  ($s) 
and  I,d7n(®s)'2  in  the  preceding  equations  shall  also  be  con. 
stant  for  all  of  these  positions.  But  even  were  these  condi- 
tions satisfied,  it  can  be  shown  that  rigorously  speaking  a 
machine  starting  from  a  state  of  rest  will  attain  a  uniform 
velocity  only  in  a  time  infinitely  great.  This  will  appear 
from  geometrical  considerations  of  a  very  simple  character, 
or  from  the  form  taken  by  equation.  By  the  first  method, 
_  let  OT,  OY  be  two  co-ordinate 

axes,  along  the  one  set  off  the 
abscissas  Qt'^  O£",  &c.?  to  re- 
present the  times  elapsed  from 
the  commencement  of  the  mo- 
tion, and  the  ordinates  £V,  t"v', 
-f —  &c.,  the  corresponding  veloci- 
ties, the  curve  Qv'v",  &c.,  will 
give  the  relation  between  the  times  and  the  velocities.  Now, 
from  the  circumstances  of  the  motion,  the  increments  of  the 
velocities  will  continually  decrease,  and  the  curve,  from  the 
law  of  continuity,  will  approach  more  nearly  to  a  right  line 


614:  EDITOEIAL    APPENDIX. 

as  the  time  increases  ;  having  for  its  assymptote  a  right  line 
parallel  to  OT,  drawn  at  a  distance  Ov  from  it,  which  is  the 
limit  the  velocity  attains  when  the  motion  becomes  uniform. 
We  moreover  see  from  the  form  the  curve  may  assume  that 
this  limit  will  be  approached  more  or  less  rapidly. 

From    (equation   B'),    representing    by   c   the    quantity 
we  obtain 

#*  dv*  -   vT>  ^L        V-P  ^       P  ^ 

-°~~      l        '      *          r** 


Now,  from  the  preceding  discussion,  the  forces  being  sup- 
posed to  act  continuously,  and  with  a  constant  intensity  and 

7Q  70 

direction,  and  the  quantities  —  -1  ,     -^-2  being  constant,  the 

ds  '     ds 

function  expressed  by  the  second  member  -of  this  equation 
has  its  greatest  value  when  ^2—  0,  or  when  the  machine  is 
about  to  move,  and  that  after  motion  begins  it  decreases 
more  or  less  rapidly  as  the  velocity  increases,  until  it  be- 
comes 0  for  a  certain  finite  value  of  the  velocity.  Hence  it 
follows  that  the  function  must  be  of  the  following,  or  some 
equivalent  form, 


in  which  ~k  is  essentially  positive  and  a  function  of  t>2  and 
certain  constants,  and  V  is  the  limit  of  the  velocity  in  ques- 
tion. We  shall  therefore  obtain  from  (equation  B'),  by  sub- 
stituting this  function  for  the  second  member, 

v 

dvz  r      cdv^ 

c  ~dt       ^v^Vty  5   and  t— J  kTV—v  Y  ' 


The  second  member  of  this  last  equation,  when  integrated 
between  the  limits  v9=Q,  arid  09=  y. ,  must  contain,  according 
to  the  known  rules  applicable  to  it,  at  least  one  term  of  the 
form  of  —  a  log.  (V—vt)  if  the  exponent  n  is  odd ;  or 
— a(Y  — /y2)~w+1,  if  n  is  even;  either  of  which  functions  will 
become  infinite  for  V— v^= 0,  or  when  v2  attains  its  limit. 

From  the  conditions  requisite  to  attain  uniformity  of  mo- 
tion in  a  machine,  the  advantages  attendant  upon  it,  so  far 
as  it  aifects  the  mechanism  are  self-apparent ;  not  only  will 
there  be  none  of  that  jarring  which  attends  abrupt  transi- 
tions in  the  velocity,  but,  from  the  manner  in  which  the 


EDITOEIAL    APPENDIX.  615 

forces  act,  the  strains  on  all  the  parts-  will  be  equable,  and 
the  respective  form  and  strength  of  each  can  thus  be  regu- 
lated in  accordance  with  the  strain  to  be  brought  upon  it, 
thus  reducing  the  bulk  and  weight  of  each  to  what  is  strictly 
requisite  for  the  safety  of  the  machine.  But  advantages  not 
less  important  than  these  result  from  the  use  of  mecnanism 
susceptible  of  uniform  motion,  owing  to  the  fact  that  for 
each  receiver  and  operator  there  is  a  velocity  for  the  applied 
and  working  points  with  which  the  functions  of  the  machine 
are  best  performed  as  respects  the  products ;  and  these 
respective  velocities  can  be  readily  secured  in  uniform 
motion  by  a  suitable  arrangement  of  the  mechanism  inter- 
mediate between  these  two  pieces. 

The  advantages  resulting  from  uniform  motion  in  machines 
has  led  to  the  abandonment  of  mechanism  that  necessarily 
causes  irregularity  of  motion,  in  many  processes  where  the 
character  of  the  operation  admits  of  its  being  done ;  and 
where,  from  the  manner  in  which  the  motive  power  acts  on 
the  receiver  and  is  transmitted  to  the  operator,  parts  with  a 
reciprocating  motion  have  to  be  introduced,  every  possible 
care  is  taken  to  so  regulate  the  action  of  these  parts  and  to 
confine  the  working  velocity  within  the  narrowest  limits  that 
the  character  of  the  operation  may  seem  to  demand.  Many 
ingenious  contrivances  have  been  resorted  to  for  this  pur- 
pose, but  as  they  belong  to  the  descriptive  part  of  mechanism 
rather  than  to  the  object  of  this  discussion,  and,  to  be  under- 
stood, would  require  diagrams  and  explanations  beyond  the 
limits  of  this  work,  they  can  only  be  here  alluded  to.  There 
is  one  however  of  general  application,  the  fly  wheel,  the 
general  theory  and  application  of  which  to  one  of  the  sim- 
plest cases  of  irregularity  are  given  in  (Arts.  75,  76,  265,  &c.) 
The  functions  of  this  piece  are  to  confine  the  change  of  velo- 
city, arising  from  irregularities  caused  either  by  the  mechan- 
ism, or  the  mode  of  action  of  the  motive  power  within  certain 
limits ;  absorbing,  by  the  resistance  offered  by  its  inertia, 
or  accumulating  work  whilst  the  motion  is  accelerated,  and 
the  work  of  the  motive  power  is  therefore  greater  than  that 
of  the  other  resistances,  and  then  yielding  it  w^hen  the  reverse 
obtains ;  thus  performing  in  machinery  like  functions  to 
those  of  regulating  reservoirs  in  the  distribution  of  water. 
It  should  however  not  be  lost  sight  of  that  whatever  resources 
the  fly  wheel  may  offer  in  this  respect  they  are  accompanied 
with  drawbacks,  inasmuch  as  the  weight  of  the  wheel,  its 
bulk  and  the  great  velocity  with  which  it  is  frequently 
required  to  revolve,  add  considerably  to  the  prejudicial 


616  EDITOKIAL    APPENDIX. 

resistances,  as  friction  and  the  resistance  of  the  air,  and  thug 
cause  a  useless  consumption  of  a  portion  of  the  work  of  the 
motive  power.  Whenever  therefore,  by  a  proper  adjustment 
of  the  motive  power  and  the  resistances,  and  a  suitable 
arrangement  of  the  mechanism,  a  sufficient  degree  of  regu- 
larity can  be  attained  for  the  character  of  the  operation,  the 
use  of  a  fly  wheel  would  be  injudicious.  In  cases  also  where, 
from  the  functions  of  the  machine,  its  velocity  is  at  times 
rapidly  diminished,  or  sudden  stoppages  are  requisite,  the 
fly  wheel  might  endanger  the  safety  of  the  machine,  or  be 
liable  itself  to  rupture,  it  should  either  be  left  out,  or  else 
the  mass  of  the  material  should  be  concentrated  as  near  as 
practicable  around  the  axis  of  rotation  ;  thus  supplying  the 
requisite  energy  of  the  fly  wheel  by  an  augmentation  of  its 
mass.  In  all  other  cases  the  matter  should  be  thrown  as  far 
from  the  axis  as  safety  will  permit,  as  the  same  end  will  be 
attained  with  less  augmentation  of  the  prejudicial  resist- 
ances. 

From  this  general  discussion  some  idea  may  be  gathered 
of  the  relations  between  the  work  of  the  power  and  that  of 
the  resistances  in  machines,  and  of  the  means  by  which  the 
latter  may  be  so  reduced  as  to  secure  the  greatest  amount  of 
the  former  being  converted  into  useful  work.  It  must  not 
however  be  concealed  that  the  problem,  as  a  practical  one, 
presents  considerable  difficulty,  and  requires,  for  its  satis- 
factory solution,  a  knowledge  of  the  various  operators  and 
receivers  of  power,  as  to  their  forms  and  the  best  modes  of 
their  action.  This  knowledge  it  is  hardly  necessary  to 
observe  must,  for  the  most  part,  be  the  result  of  experiment; 
theory  serving  to  point  out  the  best  roads  for  the  experi- 
menter to  follow.  Both  of  these  have  shown  that  the  work 
of  the  motive  power  consumed  by  the  resistances,  caused  by 
the  parts  through  which  motion  is  communicated  from  the 
receiver  to  the  operator,  is  but  a  small  fractional  part  of  the 
total  work  uselessly  consumed,  whenever  the  mechanism  has 
been  arranged  with  proper  attention  to  the  functions  required 
of  it ;  but  that  the  principal  loss  takes  place  at  the  receiver 
and  operator,  and  this  is  owing  to  the  difficulty  of  so  arrang- 
ing the  receiver  that  the  motive  power  shall  expend  upon  it 
all  its  work  without  loss  from  any  cause  ;  and  in  like  manner 
of  causing  the  operator  to  act  in  the  most  advantageous  way 
upon  the  resistance  opposed  to  it.  Some  of  the  general  con- 
ditions to  which  these  two  pieces  must  be  subjected,  as  to 
uniformity  and  continuity  of  action  of  the  motive  power  and 
the  resistances,  and  the  avoidance  of  jarring  and  shocks  have 


EDITORIAL   APPENDIX. 


617 


been  pointed  out.  as  well  as  the  fact  that  to  each  corresponds 
a  certain  velocity  by  which  the  greatest  amount  of  useful 
effect  will  be  attained. 

This  discussion  will  make  apparent  that,  comparatively 
speaking,  but  a  small  amount  of  the  work  due  to  the  motive 
power  is  expended  on  the  useful  resistance,  or  the  matter  to 
be  operated  on.  In  some  of  the  best  contrived  receivers,  as 
the  water  wheel,  for  example,  where  the  motive  power  can 
be  made  to  act  with  the  greatest  regularity,  and  the  receiver 
be  brought  to  as  near  an  approach  to  uniformity  of  motion 
as  attainable,  the  quantity  of  work  it  is  capable  of  yielding 
seldom  exceeds  eight  tenths  of  that  due  to  what  the  water 
expends  upon  it,  under  the  most  careful  arrangement  of  the 
wheel  and  the  velocity  of  its  motion. 


As  an  example  under  this  head  (Art.  149)  equation  (115), 
and  an  illustration  of  the  circumstances  attending  the  attain- 
ment of  uniformity  of  motion  Note  (t)  in  machines  ;  suppose 
the  axle  A  carrying  two  arms  B,  B,  to  the 
extremities  of  which  two  thin  rectangular 
disks  C,  C,  are  attached,  their  planes  pass- 
ing through  the  axis  of  rotation,  to  be  put 
in  motion  by  the  descent  of  a  weight  P,  at- 
tached to  a  cord  wound  round  the  axle. 
In  this  case  the  resistances  to  the  moving 
force  during  the  acceleration  will  be 
that  of  the  air  acting  against  the  disks  and 
the  two  arms,  the  inertia  of  the  parts  in 
motion,  and  the  friction  on  the  gudgeons 
of  the  axle. 

Represent  by  A  the  sum  of  the  areas  of 
the  two  disks,  a  the  distance  of  their  centres  from  the  axis, 
dm  an  elementary  mass  of  the  machine  at  the  distance  r 
from  the  axis,  w  the  angular  velocity  of  the  system,  a1  the 
radius  of  the  axle  measured  to  the  axis  of  the  cord,  p  the 
radius  of  the  gudgeon,  9  the  limiting  angle  of  resistance, 
lv  the  total  length  of  the  cord,  I  the  length  of  the  part 
unwound,  w  the  weight  of  the  unit  in  length  of  the  cord,  W 
the  total  weight  of  the  machine  excepting  Pr 

From  experiment  we  have  for  the  resistance  of  the  air  to 
the  motion  of  the  two  discs  cA-y2— cAw2<22,  in  which  v=d)O> 


618  EDITOEIAL   APPENDIX. 

expresses  the  velocity  of  the  centre  of  the  disk  and  c  a  con- 
stant determined  by  experiment.  The  resistance  offered  by 
the  inertia  of  dm  during  the  acceleration  of  the  motion  is 

represented    (Art.  95)  equations  (Y2)  (73)  by  dmr  —  ^-,  in 

Cut 

which      ^   is  the  acceleration  of  the  angular  velocity  in  the 

dt 

element  of  time  dt,  the  resistances  offered  by  the  inertia  of 
the  weight  Px  and  that  of  the  pendant  portion  of  the  cord 

represented  by  wl  are,  in  like  manner,  expressed  -i"1"^  al    ***  , 

g  dt 

the  total  pressure  upon  the  gudgeons  will  evidently  be  ex- 

pressed by  Pj  +  W-  —  l—  -  0,1  —=-,  since,  during  the  accele- 

9  dt 

ration  of  the  motion,  the  resistance  of  the  inertia  of  the 
weights  Pj  and  wl  act  in  an  opposite  direction  to  these 
weights. 

In  the  state  bordering  upon  motion  at  each  instant  there 
obtains 


(-p   .  w 
r  x  4-  W  — 


dt          g          dt 

du  \ 

a,  ---  I  p  sm.  9. 
g  dt  J 


Representing  by  n^  the  coefficient  of  wa,  by  m2  that  of  —  1 

dt 

and  by  <f  the  algebraic  sum  of  the  other  terms,  there  obtains 


dt 


•<i.i) 

From  this  last  equation  we  se  that  w  approaches  rapidly  the 

limit  i  which  it  only  attains  when  fcoo  .     As  this  limit  cor- 
n 

responds  to  that  in  which  the  motion  would  become  uni- 


EDITORIAL   APPENDIX.  619 

form,  it  might  have  been  deduced  directly  from  .the  first  of 

these  equations;  for  when  w  becomes  constant — — =0, 

cLt 


(v). 

Manner  of  estimating  the  amount  of  work  consumed  Tyy  the 
trip  hammer. 

The  trip  hammer  is  used  in  forging  heavy  iron  work, 
motion  being  given  to  it  for  this  purpose  by  teeth,  termed 


cams,  hrmly  fixed  in  an  axle  A,  termed  the  cam  shaft, 
around  which  they  are  arranged  at  equal  intervals  apart. 
The  tail  of  the  hammer  is  furnished  with  an  iron  band,  the 
upper  surface  of  which  receives  a  suitable  form  to  work 
truly  with  the  surface  of  the  cam  whilst  the  two  remain  in 
contact  during  the  ascent  of  the  head  of  the  hammer,  on  the 
same  principle  as  the  teeth  are  fashioned  in  other  cases. 
The  interval  between  the  cams  is  so  calculated  that  each 
cam  shall  take  the  band  at  rest  at  the  point  t  on  the  hori- 
zontal line  C2C,  joining  the  centres  of  rotation  of  the  cam 
shaft  and  hammer. 

To  estimate  the  work  consumed  in  the  play  of  this  ma- 
chine, it  must  be  observed  that  it  consists  of  three  distinct 
parts ;  the  first  is  that  consumed  by  the  impact  or  shock  ; 
the  second  that  due  to  the  period  after  the  shock,  in  which 
the  cam  and  tail  of  the  hammer  remain  in  contact ;  the  third 
that  consumed  by  the  cam  shaft  in  the  interval  between  the 
separation  of  the  cam  and  hammer  and  the  moment  when 
the  succeeding  cam  takes  the  hammer. 

Denote  by  R2  the  radius  of  the  primitive  circle  C2£  of  the 
cams ;  by  «2  the  angular  velocity  of  the  cam  shaft  at  any 
period  of  the  shock;  by  pa  the  radius  of  the  gudgeon  on 


620  EDITOKIAL   APPENDIX. 

which  the  shaft  revolves  ;  by  <pa  the  limiting  angle  of  resist- 
ance for  the  surfaces  of  the  gudgeon  and  its  bed  ;  by  m  an 
elementary  mass  of  the  shaft  ;  by  r  the  distance  of  m  from 
Ca  ;  by  R^C/,  wl5  pl5  px  ml9  and  TI  the  corresponding  quan- 
tities for  the  hammer. 

Now  if  we  represent  by  P  the  mutual  pressure  between 
the  surfaces  of  the  cam  and  band  at  any  period  of  the  impact, 
there  must  be  an  equilibrium  at  each  instant  between  P  and 
the  forces  of  inertia  and  the  passive  resistances  developed 
in  the  play  of  the  machine.  Considering  the  equilibrium 
around  the  axis  of  rotation  Cx  of  the  hammer  in  the  first 
place,  we  have  for  the  velocity  of  any  element  rat,  at  any 
instant,  7-^,  and  for  the  increment  of  velocity  impressed 
upon  it  by  the  cam  rjlul  ;  the  force  of  inertia  therefore  deve- 
loped by  this  increment  is  expressed  by 


and  its  moment  with  respect  to  the  axis  Ql  is 


and  the  sum  of  the  moments  of  all  the  forces  of  inertia  is 
(Arts.  95,  106) 


To  obtain  the  friction  on  the  trunnions  of  the  hammer  due 

to  P  and  the  resultant  of  the  forces  of  inertia  m.r.—^.  we 

at 

have  for  the  resultant  of  the  latter  (Art.  108)  equation  (82) 


in  which  M  represents  the  mass  of  the  hammer,  its  handle, 
&c.,  and  G  the  distance  of  its  centre  of  gravity  from  Cj  the 
axis  of  rotation.  Now,  decomposing  this  resultant  into  two 
components  perpendicular  and  parallel  to  the  line  CaCI  repre- 
senting by  a  the  angle  between  this  line  and  the  one  C,G 
through  the  centre  of  gravity  of  the  hammer,  &c.,  we  have 
for  the  perpendicular  component 


EDITORIAL   APPENDIX.  621 


cos.  a 
at 


and  for  the  parallel  one 


sn.  #. 


The  total  pressure  on  the  trunnions,  from  P  and  the  forces 
of  inertia,  will  therefore  be 


As  however,  in  most  cases  of  practice,  the  angle  a  is  either  0, 
or  very  small,  the  value  of  the  quantity  under  the  radical 
may  be  taken  without  sensible  error 


The  equation  of  equilibrium  about  the  axis  Gl  is  therefore 

^' 


(P  +  -'-MG)Pl  sin.  ?, 


. 
~  dt  '     BI-p1sin.<pl 

Now  with  respect  to  the  cam  shaft  we  have,  to  express  the 
sum  of  the  moments  of  the  forces  of  inertia  with  respect  to 
the  axis  Ca, 


As  the  pressure  on  the  trunnions  of  this  shaft  is  due  to  the 
force  P  alone,  the  moment  of  the  friction  on  them  will  be 
expressed  by  P  p  sin.  <p2. 

The  equation  of  equilibrium  of  all  the  forces  with  respect 
to  C2  will  therefore  be 


.  sin.  <(,,....  (B). 
Eliminating  P  between  equations  (A)  and  (B)  there  obtains 


622  EDITORIAL   APPENDIX. 

.^         .  :  .  .(0) 


Hie  coefficient  of  —  can  be  written  as  follows, 
dt 


placing  K  for  the  coefficient  of  MjRjRj.     Making  these  sub- 
stitutions in  equation  (C)  there  obtains 


in  which  ft  represents  the  greatest,  and  w2  the  least  angular 

velocity  of  the  cam  shaft  ;  and  w1=0,  6)1=  ^-^j  the  angu- 

-"i 

lar  velocities  of  the  hammer  ;  since  before  the  impact  it  is 
at  rest,  and  finally  attains  the  same  velocity  as  the  cam  has, 
in  which,  from  the  circumstance  of  the  mechanism,  w1R1= 
«JL 

From  the  preceding  equation  there  obtains 

"*=  ^ 


Now,  as  a  general  rule,  the  quantities  p2  sin'  ^  pl  sm>  (pl  and 

R2          Rj 

P-"S1M-b  «  —  are  ver^  sma^  W^tn  Aspect  to  unity,  and  may 


EDITORIAL    APPENDIX.  Gl'3 

therefore  be  disregarded,  and'the  quantity  K  will  differ  but 
very  little  from  unity  also.  From  this  it  will  be  seen  that 
w2  will  differ  the  less  from  12  as  M?  is  greater  than  M^  But, 
as  the  mass  of  the  cam  shaft  ordinarily  very  much  exceeds 
that  of  the  hammer,  wre  can  assume,  without  liability  to  any 
great  error,  that  the  mean  angular  velocity  of  the  cam  shaft. 
deduced  from  observing  the  number  of  revolutions  made  by 
it  in  a  given  time,  is  sensibly  the  arithmetical  mean  of  12 

and  w2.  Designating  this  mean  by  12,  we  have  Qt=  +  G)\ 
From  this  relation  and  equation  (D)  there  obtains 

0_2a.(M.+gM.).      dfc)          20.M. 
2M,+KM,  "'-2M.+  KM,- 

From  these  two  relations  the  living  force  destroyed  by  the 
impact  can  be  deduced  as  follows.  Before  the  impact  the 
living  force  of  the  cam  shaft  was  122M2RQ  ;  after  the  impact, 
as  the  point  of  contact  of  the  cam  and  band  moved  with  the 
same  velocity,  the  living  force  of  the  whole  machine  is 


The  living  force  destroyed  therefore  is  expressed  by 

a'M.B.'-u.'I^OM.+M,); 
or,  substituting  for  w2  from  equation  (D),  by 


finally,  substituting  for  12  and  wa  their  values  in  12,,  there 
obtains 


It  is  now  readily  seen,  from  the  form  of  this  last  expression 
for  the  loss  of  living  force  by  the  impact,  that,  since  K  may 


624:  EDITORIAL   APPENDIX. 

be  assumed  as  sensibly  equal  to  unity,  the  numerical  value 
of  this  expression  will  depend  upon  the  ratio  —  —  L.     Taking 


M^M,  the  value  of  the  expression  becomes  fC^MjB,*;  and 
for  Ma=oo  it  becomes  i2j2M,R22.  Therefore  between  these 
limits  the  difference  is  |  only  of  the  living  force  lost  under 
the  supposition  of  M2=o>  . 

In  the  ordinary  arrangement  of  this  machine  it  rarely 
occurs  that  M,  is  not  less  than  TVM2.  Assuming  this  as  the 
limit,  and  substituting  in  the  preceding  expression  IDIM^  for 
M2,  there  obtains  for  the  required  loss  of  living  force 
O'OTT^VMjIV.  It  is  therefore  seen  that,  in  all  usual  cases, 
M2  may  be  assumed  as  infinite  without  causing  any  notice- 
able error  in  the  result. 

To  estimate  the  accumulated  work  expended  by  the  cam 
shaft  for  each  shock,  fi,  a>2  and  fi,  being  the  same  as  in  the 
preceding  expression,  this  work  is  expressed  by 


As  the  cam  shaft  expends  this  amount  of  accumulated  work 
at  each  impact,  a  quantity  of  work  equal  to  the  half  of  this 
must  be  yielded  by  the  motive  power  at  each  impact,  or 


If  therefore  there  are  N  cams  on  the  shaft,  and  it  makes  n 
revolutions  in  one  minute,  then  the  work  consumed  by  the 
number  of  shocks  in  one  second  will  be  expressed  by 

Nn  2ni2MJVI1R22K 
60  '••'  ™ 


This  then  is  the  work  consumed  by  the  impact  in  one  second 
fov  the  first  period  of  the  play  of  the  machine  ;  and  it  has 
been  calculated  according  to  what  was  laid  down  in  Note  (t) 
on  the  subject  of  shocks,  by  disregarding  the  work  of  the 
other  forces  as  inappreciable  during  the  short  interval  of  the 
impact. 

To  estimate  now  the  work  expended  during  the  second 
period,  or  whilst  the  cam  and  band  are  in  contact  after  the 
,  let  C1G1  be  any  position  of  the  line  C,G,  during  this 


EDITORIAL    APPENDIX.  625 


period,  making  an  angle  G^G^a  with  its  position  when 
the  hammer  is  at  rest.  Represent  by  Pj  the  normal  pressure 
at  the  surface  of  contact  of  the  cam  and  band  which  will 
balance  all  the  resistances  developed  in  the  motion  of  the 
hammer,  leaving  out  of  consideration  that  of  inertia,  as  the 
change  of  velocity  between  the  end  of  the  impact  and  when 
the  cam  disengages  from  the  band  is  so  small  that  the  living 
force  due  to  this  interval  may  be  neglected  in  comparison 
with  the  work  of  the  other  forces  ;  by  Wx  the  weight  of  the 
hammer,  its  handle,  &c. 

When  the  line  GCt  is  in  the  position  G^,  the  line  Ctt  will 
oe  in  that  C^  making  the  angle  £0^—  a  with  its  original 
position.  The  force  Pj  acting  at  ^  in  this  position  and  per- 
pendicular to  the  line  tlCl  —  since  the  surface  of  the  band 
produced  passes  through  the  axis  Cn  the  surface  of  the  cam 
being  an  epicycloid  —  has  for  its  vertical  and  horizontal  com- 
ponents P,  cos.  a  and  Px  sin.  a.  The  pressure  on  the  trun- 
nions of  the  hammer,  which  is  the  resultant  of  Pl  and  W^ 
therefore  will  be  expressed  by 


,  +  Px  cos.  a)2  +  Pt*  sin.2  a  ; 

and  since  the  first  term  of  the  radical  is  in  all  cases  greater 
than  the  second,  the  value  of  the  radical  itself  Haav,  be 
expressed  by  (KOTE  B) 


!  cos.  a  +      4  sn.  a. 

The  equation  of  equilibrium  between  ~Pl  and  the  other  forces^ 
will  therefore  be 

PjB^WjGcos.  («+«)+  {/(W.+P^os.^H-^sin.^psin.^. 

The  moment  of  the  friction  at  the  point  £15  due.-to  Px  with 
respect  to  the  point  C15  in  this  case  from  the  form  .of  the  cam 
and  band,  being  0. 

As  the  pressure  P,  varies  with  the  angle  a,  we  can  -,  only 
obtain  its  mean  value  by  first  finding  its  quantity  of  work 
for  the  angle  OL=OLI  described  whilst  the  cam  and  band  are  an 
contact.  Multiplying  the  last  equation  by  da,  and  then 
integrating  between  a=0  and  a—  ^  there  obtains 


cos.  a  +      m  ?1  sn. 
40 


62G  EDITORIAL    APPENDIX. 

representing  by  POT  the  mean  value  of  P,  or  tne  constant 
force  applied  vertically  at  tf,  which  multiplied  by  !£,»„  the 
path  described  by  the  point  of  application,  will  give  the 
amount  of  work  of  the  variable  pressure  R  for  the  same 
path  ;  and  introducing  this  mean  value  into  the  term  of  the 
preceding  equation  that  represents  the  moment  of  the  fric- 
tion on  the  trunnions,  as  this  will  not  produce  any  sensible 
error  in  the  results. 

Now  observing  that  the  quantity  G  jsin.  (#  +  a)—  sin.  aj  is 
the  vertical  height  through  which  the  centre  of  gravity  of 
the  hammer,  &c.  is  raised  during  the  period  in  question,  and 
that  PmR^  is  the  work  of  the  mean  force  ;  calling  this  ver- 
tical height  A,  and  substituting  the  work  of  the  mean  for 
that  of  the  variable  force  in  the  last  equation  ;  there  obtains 


m  sn.  ai-m  cos.  a, 
Pl  sin.  <p. 


^  —  \y  sin.  c^  +  fi  (1  —  cos.a,)  \  ^  sin.  <p, 


^     ^  /™ 
' 


If  we  now  multiply  the  second  member  of  equation  (E)  by 
Rjttj  we  shall  obtain  the  approximate  value  of  the  work  of 
the  variable  force  Pl  during  the  period  in  question ;  or  the 
value  of  PmH1a1  as  determined  from  equation  (E). 

To  find  now  the  work  that  the  motive  power  must  supply 
to  the  cam  shaft  for  this  expenditure  P^R^-due  to  the 
motion  given  to  the  hammer  during  the  period  in  question, 
and  also  that/ arising  from  the  resistances  developed  by  the 
motion  of  the  cam  shaft  itself  during  this  period,  represent 
by  P3  a  force  which,  acting  at  a  distance  R3  from  the  axis  C2 
of  the  cam  shaft,  will  balance  all  the  resistances  around  C2 ; 
by  W2  the  weight  of  the  cam  shaft  and  its  fixtures  :  by  6  any 
angle  described  by  the  cam  shaft  during  the  period  con- 
sidered ;  and  9  the  limiting  angle  of  resistance  at  the  point 
of  contact  of  the  cam  and  band. 

The  pressure  on  the  tnmnions  of  the  cam  shaft  is  evidently 
expressed  by 

W2-fP3-P™; 

and  the  equation  that  expresses  the  work  of  P8  for  the  ele- 
mentary angle  dO  is 


EDITORIAL    APPENDIX.  627 

tan. 


Now  representing  by  PM  the  mean  value  of  Ps,  and  substi- 
tuting it  for  P8  in  the  last  term  of  the  second  member  of  thia 
equation,  which  may  be  done  without  causing  any  sensible 
error  in  the  result;  observing,  from  the  conditions  of  the 
mechanism  that  R^^E^  ;  and  integrating  this  equation 


between  the  limits  0  =  0  and  0=^—  _^b.;  there  obtains,  to 

Ka 

express  the  total  work  of  P3  for  the  angle  015 


*  Omitting  the  work  consumed  by  the  friction  of  the  axles  in  equation  (251) 
(Art.  220),  that  which  is  expended  on  the  teeth  in  contact  whilst  the  arc  r^  is 
described  is  represented  by  the  term  of  the  equation 


Now  if  we  suppose  a2=ra,  or  that  P2  acts  at  the  point  of  contact  and  normal 
to  the  surface,  this  term,  modified  to  suit  the  supposition,  becomes 

tan.  0= 


2 
Dividing  this  last  expression  by  r2i/>,  there  obtains, 

P2  (ri+r*\  r^L  tan.  4> 


as  the  value  of  a  mean  or  constant  force  which  applied  tangentially  to  the 
circumference  having  the  radius  r2  will  expend,  whilst  the  point  of  application 
describes  the  arc  r2i/>,  the  same  quantity  of  work  as  that  consumed  by  the  fric- 
tion of  the  teeth  in  contact  whilst  this  arc  is  described.  In  this  expression 
the  value  of  P2  is  less  than  the  true  value. 

The  foregoing  is  the  theorem  of  M.  Poncelet  referred  to  on  page  xii. 
Author's  Preface.  The  direct  manner  of  deducing  it  is  found  on  page  192 
Navier.  Resume  des  Lefons^  &c.  Troisieme  Partie.  Paris,  1838. 


628  EDITORIAL   APPENDIX. 


(F). 
R3-p2sm.<pa. 

The  work  therefore  that  the  motive  power  must  supply  to 
the  cam  shaft  during  this  period  is  found  by  multiplying  the 

second  member  of  equation  (F)  by  R301:=:R3-j|-i  or  the  path 

passed  over  by  the  point  of  application  of  the  mean  force  Pn 
during  this  period. 


Representing  in  like  manner,  by  -^-  the  number  of  times 

the  hammer  is  raised  per  second,  the  quantity  of  work  that 
the  motive  power  must  supply  for  this  expenditure  will  be 
expressed  by 

.(2). 


S 
60  60         8  R, 

During  the  last  period,  or  whilst  the  hammer  is  down,  the 
motive  power  will  only  have  to  supply  the  expenditure  of 
work  caused  by  the  friction  on  the  trunnions  of  the  cam 
shaft,  arising  from  the  weight  of  this  shaft  and  its  fixtures 
and  the  power  ;  any  accumulation  of  work  in  this  shaft 
during  this  period  being  neglected  as  small  in  amount. 
Representing  byj?=N  the  number  of  cams  on  the  shaft,  their 
distance  apart  on  the  primitive  circumference  whose  radius 

is  R2  is  evidently  -  —  ->  and,  as  the  arc  described  on  this 
circumference  whilst  the  cam  shaft  and  hammer  are  engaged 

is  R20j,  that  described  whilst  the  hammer  is  down  is  --  a— 

P 
RjCtj.     Calling  Pp  the  power  which  acting  at  the  distance  R, 

will  balance  the  friction  arising  from  the  weight  W2  of  the 
cam  shaft  and  fixtures  and  P2,  the  value  of  Pp  will  be  found 
according  to  the  conditions  stated  as  follows, 


The  work  of  Pp  is 


W2p2  sin.  9, 

trl 2  —  Riai) 

R,\   P  I 


EDITORIAL   APPENDIX.  629 

as  the  path  passed  over  by  its  point  of  application  is  evi- 
dentlythearc 


The  work  which  the  motive  power  must  supply  therefore 
per  second  during  this  last  period  is  expressed  by 


By  taking  the  sum  of  the  quantities  expressed  by  the 
formulas  (1),  (2),  and  (3)  there  obtains 

K3 


to  express  the  total  work  that  the  motive  power  must  yield 
to  the  cam  shaft  per  second  to  supply  the  work  consumed 
by  all  the  resistances. 

Tiiat  consumed  by  the  useful  resistances,  which  consist  of 
half  the  living  force  transmitted  to  the  hammer  and  the 
work  consumed  in  raising  the  centre  of  gravity  of  the  ham- 
mer, &c.,  through  the  vertical  height  h  is  represented  by 

o^MJV     w  .  _2ni2M,M1R '    w  , 
—  -(2M8  +  KM1)  + 

From  the  preceding  expressions,  it  is  easy  to  deduce  the 
work  which  must  be  expended  in  producing  a  given  depth 
of  indentation  by  the  hammer  upon  the  metal  when  brought 
to  a  given  state  of  heat.  For  this  purpose,  we  observe  that 
to  half  the  living  force  acquired  by  the  hammer  there  cor- 
responds a  certain  amount  of  work,  estimated  in  terms  of 
the  weight  of  the  hammer  and  a  certain  height  hl  to  which 
its  centre  of  gravity  has  been  raised,  and  expressed  by 

0)aM,  R,2          TTT     7 


2 

the  total  work  therefore  expended  by  the  hammer  in 
indenting  the  metal  is  expressed  by  W^-fW^ ;  since,  from 
the  state  of  the  metal  the  molecules  which  are  displaced  by 
the  impact  acquire  velocities  which  are  not  appreciable  from 
their  smallness ;  the  resistances  therefore  offered  by  the 
metal  to  indentation  may  be  regarded  as  independent  of  the 


630  EDITOEIAL    APPENDIX. 

velocity  and,  from  the  laws  of  the  penetration  of  solids  into 
different  media,  proportional  simply  to  the  area  of  the  inden- 
tation. Representing  then  by  a  and  ~b  the  sides  of  the  area 
of  the  indentation,  supposed  rectangular,  at  the  surface  of 
the  metal  impinged  on,  d  the  depth  of  the  indentation,  and 
C  the  constant  ratio  of  the  resistance  and  the  area  of  the 
indentation,  the  following  relation  obtains  between  the  work 
expended  by  the  hammer  in  its  fall  and  that  offered  by  the 
resistance  of  the  metal 


an  equation  from  which  C  may  be  determined  by  experi- 
ment'in  any  particular  case. 

It  will  be  readily  seen  that  the  preceding  expressions  will 
be  rendered  applicable  to  the  cases  where  the  cam  catches 
the  hammer  on  the  same  side  of  its  axis  of  rotation  as  its 

centre  of  gravity,  by  writing  —  ^L  MG  for  +  ^-MG,  and 

at  at 

moreover  in  this  case  when  P—  5_ L  MG=0,  there  will  be  no 

dt 

shock  on  the  trunnions  (Arts.  108,  109),  and  there  then 
obtains,  to  find  the  point  where  the  cam  should  catch  the 
hammer  corresponding  to  this  case, 

E=MG 

*  Morin,  Suite  des  Nouvelles  Experiences  sur  le  Frottenient,  p.  67.    Paris,  1836. 


APPENDIX. 


NOTE  A. 


THEOBEM.  —  The  definite  integral  J  fxdx  is  the  limit  of  the  sums  of  tht 

a 

values  severally  assumed  by  the  product  fx  .  Ax,  as  x  is  made  to  vary  by 
successive  equal  increments  of  Ax,  from  a  to  b,  and  as  each  such  equal 
increment  is  continually  and  infinitely  diminished,  and  their  number  there- 
fore continually  and  infinitely  increased. 

To  prove  this,  let  the  general  integral  be  represented  by  Fa;  ;  let  us  sup- 
pose that  fx  does  not  become  infinite  for  any  value  of  x  between  a  and  b, 
and  let  any  two  such  values  be  x  and  x  +  Ax  ;  therefore,  by  Taylor's  the- 
orem, F  (x  +  Ace)  =  Fx  +  Axfx  +  (Ax)  '+XM,  where  the  exponent  1  +  a,  is 
given  to  the  third  term  of  the  expansion  instead  of  the  exponent  2,  that  the 

case  may  be  included  in  which  the  second  differential  coefficient  of  Fa;,  -^—  , 

dx 

is  infinite,  and  in  which  the  exponent  of  Aa;  in  that  term  is  therefore  a 
fraction  less  than  2. 

Let  the  difference  between  a  and  b  be  divided  into  n  equal  parts  ;  and 
let  each  be  represented  by  Aa;,  so  that 

&—  a 

—  =  Aa;. 
n 

Giving  to  a;,  then,  the  successive  values  a,  a+  Aa;,  a  +  2  A  a;  .  .  a+(n  —  1) 
A  a;,  and  adding, 


.'.  F&—  Ya=Ax%l*f{a  +  (n—  l)Ax} 
Now  none  of  the  values  of  M  are  infinite,  since  for  none  of  these  values  is 
fx  infinite.  If,  therefore,  M  be  the  greatest  of  these  values,  then  is  SM,  less 
than  riM.  :  and  therefore 

F&  _  Ya  —  Aa;2  tf{a  +  (n—  1)  Aa;}  <  (o—a)  M  (Aa;)X. 

The  difference  of  the  definite  integral  F5  —  Fa,  and  the  sum  JZin(Ax)f{a-{' 
(n  —  1)  Aa;}  is  always,  therefore,  less  than  (b  —  a)  M  (A#)A.  Now  M  is  finite, 
and  (&  —  a)  is  given,  and  as  n  is  increased  Aa;  is  diminished  continually  ; 
and  therefore  (Aa;)x  is  diminished  continually,  a,  being  positive. 

Thus  by  increasing  n  indefinitely,  the  difference  of  the  definite  integral 


£32  APPENDIX. 

and  the  sum  may  be  diminished  indefinitely,  and  therefore,  in  the  limit,  the 
definite  integral  is  equal  to  the  sum  (i.  e.) 

FJ—F0  =  limit  2tB (Atf)  ./{a  +  (n— 1)  Az}  ; 

or,  interpreting  this  formula,  F5— F<z  is  the  sum  of  the  values  of  AJK  ./a?, 
when  x  is  made  to  pass  hy  infinitesimal  increments,  each  represented  by 
Afc  from  a  to  &. 


NOTE  B. 
PONOELET'S  FIEST  THEOEEM. 


*  The  values  of  a  and  &  in  the  radical  -v/^2  +  ^a  being  linear  and  rational, 
let  it  be  required  to  determine  the  values  of  two  indeterminate  quantities 
a  and  ]5,  such  that  the  errors  which  result  from  assuming  •v/aa  +  &*=  a#  +  j3&, 

f-\ 

through  a  given  range  of  the  values  of  the  ratio  (  &  ),  may  be  the  least  pos- 

sible in  reference  to  the  true  value  of  the  radical  ;  or  that  ***  +  &>•—    a*_  +  V 


or  -  .——  =1-  —  i^  may  be  the  least  possible  in  respect  to  all  that  range  of 
values  which  this  formula  may  be  made  to  assume  between  two  given 
extreme  values  of  the  ratio  T.  Let  these  extreme  values  of  the  ratio  ^ 

be  represented  by  cot.  ifo  and  cot.  ^2,  and  any  other  value  by  cot.  4.     Sub- 

d  __ 

stituting  cot.  4-  for  T  in  the  preceding  formula,  and  observing  thaty  a2-f&a 


=  -v/&2cot..24/  +  &2=  &  cosec.  4,  also  that  a«+|3&  =  a&  cot.  4/+|3&=(a  cos. 
sin.  4)&  cosec.  4,  the  corresponding  error  is  represented  by 

a  cos.  4-+  18  sin.  4—1  .....  (1); 

which  expression  is  evidently  a  maximum  for  that  value  <//3  of  i//  which  is 
determined  by  the  equation 

a 
cot.T//,=jg  .....  (2); 

so  that  its  maximum  value  is 

-V/^HS2—  1  .....  (3). 

Moreover,  the  function  admits  of  no  other  maximum  value,  nor  of  any 
minimum  value.    The  values  of  a  and  0  being  arbitrary,  let  them  be 

assumed  to  be  such  that  -  or  cot.  i//3  may  be  less  than  cot.  ^  and  greater 

*  The  method  of  this  investigation  is  not  the  same  as  that  adopted  by  M. 
Poncelet  ;  the  principle  is  the  same. 


PONCELET'S  THEOREM.  G33 

than  cot.  4/a.  Now,  so  long  as  all  the  values  of  the  error  (formula  1) 
remain  positive,  between  the  proposed  limits,  they  are  all  manifestly  di- 
minished by  diminishing  a  and  j3  ;  but  when  by  this  diminution  the  error 
is  at  length  rendered  negative  in  respect  to  one  or  both  of  the  extreme 
values  4,!,  or  4/2  of  4,  and  to  others  adjacent  to  them,  then  do  these  nega- 
tive errors  continually  increase,  as  a  and  |3  are  yet  farther  diminished, 
whilst  the  positive  maximum  error  (formula  3)  continually  diminishes. 
Now  the  most  favorable  condition,  in  respect  to  the  whole  range  of  the 
errors  between  the  proposed  limits  of  variation,  will  manifestly  be  attained 
when,  by  thus  diminishing  the  positive  and  thereby  increasing  the  negative 
errors,  the  greatest  positive  error  is  rendered  equal  to  each  of  the  two 
negative  errors;  a  condition  which  will  be  found  to  be  consistent  with 
that  before  made  in  respect  to  the  arbitrary  values  of  a  arid  j3,  and  which 
supposes  that  the  values  of  the  error  (formula  1)  corresponding  to  the 
values  4»,  and  42  are  each  equal,  when  taken  negatively,  to  the  maximum 
error  represented  by  formula  3,  or  that  the  constants  a  and  ]3  are  taken 
so  as  to  satisfy  the  two  following  equations. 


1—  (a  cos.  *,+]3  sin.  ¥,)=  V'aM-jS8—  1. 
1—  (a  cos.  •*•,+£  sin.  •*•,)=!  —  (a  cos.  V^—p  sin.  •*•„). 
The  last  equation  gives  us  by  reduction 


and  a  =  j8  cot.  £(•*•,  +  •*•„). 

Substituting  these  values  in  the  first  equation,  and  reducing, 
2  sin.  K*i+^)         sin,  j+y, 


1+008.  *(*•,  —  *,)       COS.  2i(¥,  — 
2  COS- 


1  +  cos.  i(%  —  ¥,)     cos.  H(*i  —  ¥,)  " 

These  values  of  a  and  ]3  give  for  the  maximum  error  (formula  3)  the  ex- 
pression 

tan.  '*(*!  —  *0  .....  (6). 

Thus,  then,  it  appears  that  the  value  of  the  radical  vV  +  53  is  represented, 
in  respect  to  all  those  values  of  j-  which  are  included  between  the  limits 
cot.  ¥,  and  cot.  ¥„  by  the  formula 


with  a  degree  of  approximation  which  is  determined  by  the  valu«  of 

tan.'K*.  —  ^)- 

If  in  the  proposed  radical  the  value  of  a  admits  of  being  increased  in- 
finitely in  respect  to  5,  or  the  value  of  5  infinitely  diminished  in  respect  to 
a,  then  cot.  ¥,  =  infinity  ;  therefore  ¥,  =  0.  In  this  case  the  formula  of 
approximation  becomes 


634 


APPENDIX. 


a  (I— 
and  the  maximum  error 


(9). 


If  the  values  of  a  and  &  are  wholly  unlimited,  so  that  a  may  be  infinitely 
small  or  infinitely  great  as  compared  with  &,  then  cot.  ^  =  infinity, 
cot.  ¥,  =  0  ; 


therefore  ^i=0,  ¥»=o-    Substituting  these  values,  the  formula  of  approx- 

imation becomes 

•82840+  -82845  .....  (10); 
and  the  maximum  error 

•1716,  or  |th  nearly. 

'If  &  is  essentially  less  than  a,  but  may  be  of  any  value  less  than  it,  so 
that  T  is  always  greater  than  unity,  but  may  be  infinite,  then  cot.  ^  =  in- 

finity, cot.  ^2=1  5  therefore  ^r=0,  ^2=0'     Substituting  these  values  in  the 
formula  of  approximation,  and  reducing,  it  becomes 

•960460  +  -39783&  .....  (11); 
and  the  maximum  error 

•03945,  or  ^\th  nearly. 

It  is  in  its  application  to  this  case  that  the  formula  has  been  employed  in 
the  preceding  pages  of  this  work. 

The  following  table,  calculated  by  M.  Gosselin,  contains  the  values  of 
the  coefficients  a,  and  ]3  for  a  series  of  values  of  the  inferior  limit  cot.  «£„  the 
superior  limit  being  in  every  case  infinity. 


EelationofatoJ. 

1? 

Value  of  o. 

Value  of/?. 

Maximum  Error. 

Approximate  Value 

a  and  6  any  > 
whatever    J 

0 

0-82840 

0-82840 

0-17160  or  | 

0-8284  (a  +  6) 

a>  b 

1 

0-96046 

0-39783 

0-03954  or  Jy 

•96046a  +  -397836 

o>  26 
a  >  36 

2 
3 

0-98592 
0-99350 

0-23270 
0-16123 

0-01408  or  ~!_ 
0-00650  or  TjT 

•98592a+  -232706 
•99350a+  -161236 

a>  46 

4 

0-99625 

0-12260 

0-00376  or  ^ 

•996250  +  -122606 

a>  56 

5 

0-99757 

0-09878 

0-00243  or  r}? 

•997570  +  -098786 

a>  66 

6 

0-99826 

0-08261 

0-00174  or  y]T 

•99826a  +  -082616 

i>  76 

7 

0-99875 

0-07098 

0-00125  or  ^  j^ 

•99875a  +  -070986 

a>  86 

8 

0-99905 

0-06220 

0-00095  or  T^ 

•99905a+  -062206 

a>  96 

9 

0-99930 

0-05535 

0-00070  or  -j-^ 

•99930a+  -055356 

a>  106 

10 

0-99935 

0-04984 

0-00065  or  ^^ 

•99935a+  -049846 

POWCELET'S  SECOND  THEOKEM.  635 


PONCELET'S  SECOND  THEOEEM. 

To  approximate  to  the  value  of  Va?— &2,  let  aa  —  35  be  the  formula  of 
approximation,  then  will  the  relative  error  be  represented  by 


Now,  let  it  be  observed  that  aa  being  essentially  greater  than  &*->!; 


let  j,  therefore,  be  represented  by  cosec.  4/,  then  will  the  relative  error  ba 

(a  cosec.  4/—  3) 
represented  by  1  —  —  -  ~,  or  by 

V  cosec.  "4-1 

1—  asec.4/+3tan.  4/  .....  (12), 

which  function  attains  its  maximum  when  sin.  4»  =  -.    Substituting  this 

a 

value  in  the  preceding  formula,  and  observing  that  —a  sec.  4/  +  6  tan.  4/  = 


(-9 


— sec.4*  (a— 3  sin4)=—     . — _=  —  \^L.^  we  obtain  for  the  maximum 
1- 


error  the  expression 

1—V3=&  .....  (13), 
Assuming  4/,  and  4/2  to  represent  the  values  of  4,  corresponding  to  the 

greatest  and  least  values  of  -,  and  observing  that  in  this  case,  as  in  the 

preceding,  the  values  of  a  and  3,  which  satisfy  the  conditions  of  the 
question,  are  those  which  render  the  values  of  the  error  corresponding  to 
these  limits  equal,  when  taken  with  contrary  signs,  to  the  maximum  error, 
we  have 

—  •  1  +a  sec.  4,—  3  tan.  4,  =  1  —  tV—  3a  ----  (14). 
1  —  a  sec.  4,  +  3  tan.  4,,=!  —  asec.4/,+3tan.4/,  ....  (15). 
The  latter  equation  gives,  by  reduction, 

cos.  i(4/.—  4*)  /-^ 


_  2 

~          ' 


And  a  sec.  «£,  +  /3  tan.  »//,=  0  cot.  i  (^i  +  <//,)  .  .  .  .  (17). 
Substituting  these  values  in  equation  (14),  and  solving  in  respect  to  j(? 


636  APPENDIX. 


cos.  i  (»//,  + i/>2)  +  Vcos.  »/>,  cos.  </>j 
2  cos.  i  (?//,  —  ^2) 

a  := —  «  «  i 

cos.  i  (^,  + 1^2)  +  r  cos.  i//i  cos.  ?//, 
The  maximum  error  is  represented  by  the  formula 

2  4/cos.  T//J  cos.  t//2 


These  formulaa  will  be  adapted  to  logarithmic  calculation,  if  we  assume 

d  (^t  +  ^»)="^n  and  c.08'    ;  ;  =  cosec.  ¥2 ;  we  shall  thus  obtain  from 

sin.  %  (j^i  +  t//2) 

equations  (16)  and  (17)  a  =  j3  cosec.  ^2,  Vo?  —  j32  =  j3  cot.  ^2,  and  a  sec.  ^ 
— j3  tan.  i//i  =  3  cot.  ^i ;  therefore,  by  equation  (14), 

2  2  sin. 


'  cot.  ^rl  +  cot.  ^2      sin.  (^  +  ¥a/ 

2  cosec.  ^2  2  sin.  "SP", 

cot.  ^,  +  cot.  ¥,  =  sin.  (VK1  +  ^ 


sn.      ,  — 
Max,mum  errror  = 


sn         +        _  _  _  _ 

The  form  under  which  this  theorem  has  been  given  by  M.  Poncelet  is 
different  from  the  above.     Assuming,  as  in  the  previous  case,  the  limiting 

values  of  -  to  be  represented  by  cot.  ^,  and  cot.  i/^  and  proceeding  by  a 
& 

geometrical  method  of  investigation,  he  has  shown  that  if  we  assume 
tan.  «h  =  cos.  «„  tan.  i//z  =  cos.  «2,  «!  +  w2  =  2y,,  «!  —  u>2  =  25,  and  cos.  y8  = 

;  then 


cos.  5 

2  COS.  y,  2  COS.2  y,  sin.  (y,  —  y2) 

a     —  —  -.  —  —^  &=-.  —  -,  -  r-=  -  ,  maximum  error  =  —  —  -±  --  ~. 

Sin.  (y,  +  y2y          Sin.  (y,  +  y2)  COS.  8  Sin.  (y  ,  +  y2) 

If  the  least  possible  value  of  a  be  lTlo&?  and  its  greatest  possible  value 
be  infinite  as  compared  with  5,  M.  Poncelet  has  shown  the  formula  of 
approximation  to  become 


Va?  —  &8  =  1-13190  —  0-72636& (23), 

with  a  possible  error  of  0-1319  or  |  nearly. 

If  the  least  possible  value  of  a  be  25,  and  its  greatest  possible  valua 
infinite  compared  with  5 ;  then 

4/^Zy  =  1-0186230  —  0-2T2944& (24), 

with  a  possible  error  of  -0186  or  ^d  nearly. 


ON   THE   ROLLING    OF    SHIPS.  637 

NOTE  0. 
ON  THE  ROLLING  OF  SHIPS. 

(First  published  ty  the  Author  in  the  Transactions  of  the  Royal  Society 

for  1850,  Part  //.) 

Let  a  body  be  conceived  to  float,  acted  upon  by  no  other  forces  than  its 
weight  "W,  and  the  upward  pressure  of  the  water  (equal  to  its  weight)  ; 
which  forces  may  be  conceived  to  be  applied  respectively  to  the  centre  of 
gravity  of  the  body  and  to  the  centre  of  gravity  of  the  displaced  fluid  ; 
and  let  it  be  supposed  to  be  subjected  to  the  action  of  a  third  force  whose 
direction  is  parallel  to  the  surface  of  the  fluid.  Let  AHi  represent  the  ver- 
tical displacement  of  the  centre  of  gravity  of  the  body  thereby  produced*, 
and  AH,  that  of  the  centre  of  gravity  of  its  immersed  part.  Let  more- 
over the  volume  of  the  immersed  part  be  conceived  to  remain  unaltered  f 
whilst  the  body  is  in  the  act  of  displacement.  If  each  centre  of  gravity 
be  assumed  to  ascend,  the  work  of  the  weight  of  the  body  will  be  repre- 
sented by  —  W.AH,,  and  that  of  the  upward  pressure  of  the  fluid  by  + 
W.AH2,  the  negative  sign  being  taken  in  the  former  case  because  the  force 
acts  in  a  direction  opposite  to  that  in  which  the  point  of  application  is 
moved,  and  the  positive  sign  in  the  latter,  because  it  acts  in  the  same  direc- 
tion, so  that  the  aggregate  work  2u2  (see  equation  1,  p.  122.)  of  the  forces 
which  constituted  the  equilibrium  of  the  body  in  the  state  from  which  it 
has  been  disturbed  is  represented  by 


Moreover,  the  system  put  in  motion  includes,  with  the  floating  body,  the 
particles  of  the  fluid  displaced  by  it  as  it  changes  its  position,  so  that  if 
the  weight  of  any  element  of  the  floating  body  be  represented  by  to,,  and 
of  the  fluid  by  wz,  and  if  their  velocities  be  0,  and  02,  the  whole  vis  viva  is 
represented  by 


*  "When  a  floating  body  is  so  made  to  incline  from  any  one  position  into  any 
other  as  that  the  volume  of  fluid  displaced  by  it  may  in  the  one  position  be 
equal  to  that  in  the  other,  its  centre  of  gravity  is  also  vertically  displaced  ; 
for  if  this  be  not  the  case,  the  perpendicular  distance  of  the  centre  of  gravity 
of  the  body  from  its  plane  of  flotation  must  remain  unchanged,  and  the  form 
of  that  portion  of  its  surface,  which  is  subject  to  immersion,  must  be  determined 
geometrically  by  this  condition  ;  but  by  the  supposition  the  form  of  the  body 
is  undetermined.  It  is  remarkable  what  currency  has  been  given  to  the  error, 
that  whilst  a  vessel  is  rolling  or  pitching,  its  centre  of  gravity  remains  at  rest 
I  should  not  otherwise  have  thought  this  note  necessary. 

f  This  supposition  is  only  approximately  true. 

\  If  the  centre  of  gravity  of  the  body  or  of  the  displaced  fluid  descends  (» 
property  which  will  be  found  to  characterise  a  large  class  of  vessels),  AH,  in 
the  one  case,  and  ^ILj  in  the  other,  will  of  course  take  the  negative  sign. 


t)38  APPENDIX. 

~2w,f>?  +  -5«Wi> 
and  we  have  by  equation  1  (p.  122), 

—  W(AH»—  AH^=S«I«!  +      Sw1«;  ....  (25). 


In  the  extreme  position  into  which  the  body  is  made  to  roll  and  IE 
which  Xw,flJ=0, 


or  if  the  inertia  of  the  displaced  fluid  be  neglected, 

U(5)=W.(AH1  —  AH2)  .....  (27). 

Whence  it  follows  that  the  work  necessary  to  incline  a  floating  body 
through  any  given  angle  is  equal  to  that  necessary  to  raise  it  bodily  through 
a  height  equal  to  the  difference  of  the  vertical  displacements  of  its  centre 
of  gravity  and  of  that  of  its  immersed  part  ;  so  that  other  things  being 
the  same,  that  ship  is  the  most  stable  the  product  of  whose  weight  by  this 
difference  is  the  greatest. 

In  the  case  in  which  the  centre  of  gravity  of  the  displaced  fluid  descends, 
the  sum  of  the  displacements  is  to  be  taken  instead  of  the  difference. 

This  conclusion  is  nevertheless  in  error  in  the  following  respects  :  — 

1st.  It  supposes  that  throughout  the  motion  the  weight  of  the  displaced 
fluid  remains  equal  to  that  of  the  floating  body,  which  equality  cannot 
accurately  have  been  preserved  by  reason  of  the  inertia  of  the  body  and 
of  the  displaced  fluid.* 

From  this  cause  there  cannot  but  result  small  vertical  oscillations  of  the 
body  about  those  positions  which,  whilst  it  is  in  the  act  of  inclining,  cor- 
respond to  this  equality,  which  oscillations  are  independent  of  its  principal 
oscillation. 

2ndly.  It  involves  the  hypothesis  of  absolute  rigidity  in  the  floating 
body,  so  that  the  motion  of  every  part  and  its  vis  viva  may  cease  at  once 
when  the  principal  oscillation  terminates.  The  frame  of  a  ship  and  its 
masts  are,  however,  elastic,  and  by  reason  of  this  elasticity  there  cannot 


*  The  motion  of  the  centre  of  gravity  of  the  body  being  the  same  as  though 
all  the  disturbing  forces  were  applied  directly  to  it,  it  follows,  that  no  elevation 
of  this  point  is  caused  in  the  beginning  of  the  motion,  by  the  application  of  a 
horizontal  disturbing  force,  or  by  a  horizontal  displacement  of  the  weight  of 
the  body,  which,  if  it  be  a  ship,  may  be  effected  by  moving  .its  ballast.  The 
motion  of  rotation  thereby  produced  takes  place  therefore,  in  the  first  instance, 
about  the  centre  of  gravity,  but  it  cannot  so  take  place  without  destroying  the 
equality  of  the  weight  of  the  displaced  fluid  to  that  of  the  body.  From  this 
inequality  there  results  a  vertical  motion  of  the  centre  of  gravity,  and  anothei 
axis  of  rotation. 


ON   THE   ROLLING   OF   SHIPS.  639 

but  result  oscillations,  which  are  independent  of,  and  may  not  synchro- 
nise with,  the  principal  oscillation  of  the  ship  as  she  rolls,  so  that  the  VIA 
viva  of  every  part  cannot  be  assumed  to  cease  and  determine  at  one  and 
the  same  instant,  as  it  has  been  supposed  to  do. 

3rdly.  No  account  has  been  taken  of  the  work  expended  in  communi- 
cating motion  to  the  displaced  fluid,  measured  by  half  its  vis  viva  and 

represented  by  the  term  —  ^w^vl  in  equation  26. 


From  a  careful  consideration  of  these  causes  of  error,  the  author  was 
led  to  conclude  that  they  would  not  affect  that  practical  application  of  the 
formula  which  he  had  principally  in  view  in  investigating  it,  especially  as 
in  certain  respects  they  tended  to  neutralise  one  another.  The  question 
appeared,  however,  of  sufficient  importance  to  be  subjected  to  the  test  of 
experiment,  and  on  his  application,  the  Lords  Commissioners  of  the  Admi- 
ralty were  pleased  to  direct  that  such  experiments  should  be  made  in  Her 
Majesty's  Dockyard  at  Portsmouth,  and  Mr.  FINCHAM,  the  eminent  Master 
Shipwright  of  that  dockyard,  and  Mr.  KAWSON,  were  kind  enough  to 
undertake  them. 

These  experiments  extended  beyond  the  object  originally  contemplated 
by  him  ;  and  they  claim  to  rank  as  authentic  and  important  contributions 
to  the  science  of  naval  construction,  whether  regard  be  had  to  the  prac- 
tical importance  of  the  question  under  discussion,  the  care  and  labor 
bestowed  upon  them,  or  the  many  expedients  by  which  these  gentlemen 
succeeded  in  giving  to  them  an  accuracy  hitherto  unknown  in  experiments 
of  this  kind. 

That  it  might  be  determined  experimentally  whether  the  work  which 
must  be  done  upon  a  floating  body  to  incline  it  through  a  given  angle  be 
that  represented  by  equation  27,  it  was  necessary  to  do  upon  such  a  body 
an  amount  of  work  which  could  be  measured  ;  and  it  was  further  neces- 
sary to  ascertain  what  were  the  elevations  of  the  centres  of  gravity  of  the 
body  and  of  its  immersed  part  thus  produced,  and  then  to  see  whether 
the  amount  of  work  done  upon  the  body  equalled  the  difference  of  these 
elevations  multiplied  by  its  weight. 

To  effect  this,  the  author  proposed  that  a  vessel  should  be  constructed 
of  a  simple  geometrical  form,  such  that  the  place  of  the  centre  of  gravity 
of  its  immersed  part  might  readily  be  determined  in  every  position  into 
which  it  might  be  inclined,  that  of  its  plane  of  flotation  being  supposed  to 
be  known  ;  and  that  a  mast  should  be  fixed  to  it,  and  a  long  yard  to  this 
mast,  and  that  when  the  body  floated  in  a  vertical  position  a  weight 
suspended  from  one  extremity  of  the  yard  should  suddenly  be  allowed  to 
act  upon  it  causing  it  to  roll  over;  that  the  position  into  which  it  thus 
rolled  should  be  ascertained,  together  with  the  corresponding  elevations 
of  its  centre  of  gravity  and  the  centre  of  gravity  of  its  immersed  part, 
and  the  vertical  descent  of  the  weight  suspended  from  the  extremity  of 
its  arm.  The  product  of  this  vertical  descent  by  the  weight  suspended 


640  APPENDIX. 

from  the  arm  ought  then,  by  the  formula,  to  be  found  nearly  equal  to  the 
difference  of  the  elevations  of  the  two  centres  of  gravity  multiplied  by 
the  weight  of  the  body ;  and  this  was  the  test  to  which  it  was  proposed 
that  the  formula  should  be  subjected,  with  a  view  to  its  adoption  by  prac- 
tical men  as  a  principle  of  naval  construction. 

To  give  to  the  deflecting  weight  that  instantaneous  action  on  the  ex- 
tremity of  the  arm  which  was  necessary  to  the  accuracy  of  the  experiment, 
a  string  was  in  the  first  place  to  be  affixed  to  it  and  attached  to  a  point 
vertically  above,  in  the  ceiling.  When  the  deflecting  weight  was  first 
applied  this  string  would  sustain  its  pressure,  but  this  might  be  thrown 
at  once  upon  the  extremity  of  the  arm  by  cutting  it.  A  transverse  sec- 
tion of  the  vessel,  with  its  mast  and  arm,  was  to  be  plotted  on  a  large 
scale  on  a  board,  and  the  extreme  position  into  which  the  vessel  rolled 
being  by  some  means  observed,  the  water-line  corresponding  to  this 
position  was  to  be  drawn.  The  position  of  the  yard,  in  respect  to  the 
surface  of  the  water  in  that  position,  would  then  be  known,  and  the  vertical 
descent  of  the  deflecting  weight  could  be  measured,  and  also  the  vertical 
ascent  of  the  centre  of  gravity  of  the  immersed  part  or  displacement. 

To  determine  the  position  of  the  centre  of  gravity  of  the  vessel,  it  was 
to  be  allowed  to  rest  in  an  inclined  position  under  the  action  of  the  deflect- 
ing weight ;  and  the  water-line  corresponding  to  this  position  being  drawn 
on  the  board,  the  corresponding  position  of  the  deflecting  weight  and  of 
the  centre  of  gravity  of  the  immersion  were  thence  to  be  determined. 
The  determination  of  the  position  of  the  vertical  passing  through  the 
centre  of  gravity  of  the  body  would  thus  become  an  elementary  question 
of  statics ;  and  the  intersection  of  this  line,  with  that  about  which  the 
section  was  symmetrical,  would  mark  the  position  of  the  centre  of  gravity. 
This  determination  might  be  verified  by  a  second  similar  experiment  with 
a  different  deflecting  weight. 

These  suggestions  received  a  great  development  at  the  hands  of  Mr. 
RAWSON,  and  he  adopted  many  new  and  ingenious  expedients  in  carrying 
them  out.  Among  these,  that  by  which  the  position  of  the  water-line 
was  determined  in  the  extreme  position  into  which  the  vessel  rolls,  is 
specially  worthy  of  observation.  A  strip  of  wood  was  fastened  at  right 
angles  to  that  extremity  of  the  yard  to  which  the  deflecting  weight  was 
attached,  of  sufficient  length  to  dip  into  the  water  when  the  vessel  rolled ; 
on  this  slip  of  wood,  and  also  on  the  side  of  the  vessel  nearest  to  it,  a 
strip  of  glazed  paper  was  fixed.  The  highest  points  at  which  these  strips 
of  paper  were  wetted  in  the  rolling  of  the  vessel,  were  obviously  points 
in  the  water  line  in  its  extreme  position,  and  being  plotted  upon  the  board, 
a  line  drawn  through  them  determined  that  position  with  a  degree  of 
accuracy  which  left  nothing  to  be  desired. 

Two  forms  of  vessels  were  used ;  one  of  them  had  a  triangular  and  the 
other  a  semicircular  section.  The  following  table  contains  the  general 
results  of  the  experiments. 


ON   THE   ROLLING    OF    SHIPS. 


641 


'f     'f! 

II 

ft|  * 

**4 

IJ1J 

L 

*  w 

si  w 

*=jj 

*l!u 

Form  of  thi- 
model  ex- 
perimented 

No.  of 
experi- 

Weieht  of 
motlr!  and 
loading. 

Disturb- 
„&«. 

i! 

:  L 
ii« 
ii< 

•3^ 

1 

IP    L 

\\*& 

-'li* 

n  in  whir 
v  rested  wh 

I-  action  of 
ght. 

iSjfl 

g  -.SB'S 

ji;ii 

ii 

Q 

l|ll 

III 

1 

%  z 

nif  ii 

wf 

sSi 

Pa! 

fill 

"  112 

*€l*s 
|lf*l 

Sill* 
fllj 

Ibs. 

Ibs. 

0             , 

0           , 

0           , 

1 

38-8626 

•5485 

•5161 

•5361 

23  80 

12  80 

•8961 

Triangular 

2. 

36-8690 

•8450 

•4887 

•4951 

15  30 

8    0 

•98114 

3 

87-8568 

•5877 

1-1724 

1-4503 

24    0 

13    0 

•88512 

4. 

88-2911 

•5789 

1-2678 

1-8460 

25    0 



13  80 

•9380 

Circular 
model. 

1. 
2. 
8- 

19T-18 
197-18 
255-43 

2-8225 
1-9570 
1-9570 

7-8761 
8-2486 
1-7727 

7-894 
8-122 
1-7667 

26    0 
17    0 
10    0 

24  20 
16  22 
10    0 

18    0 
9    0 
4  30 

In  the  experiments  with  the  smaller  triangular  model  the  differences^ 
between  the  results  and  those  given  by  the  formula  are  much  greater  tba&i 
in  the  experiments  with  the  heavier  cylindrical  vessel. 

In  explanation  of  this  difference,  it  will  be  observed,  first,  that  t.he  <aon-- 
ditions  of  the  experiment  with  the  cylindrical  model  more  nearly  approach 
to  those  which  are  assumed  in  the  formula  than  those  with,  the,  other ;  the 
disturbance  of  the  water  in  the  change  of  the  position  of  the  former  being 
less,  and  therefore  the  work  expended  upon  the  inertia  of  the  water,  of 
which  the  formula  takes  no  account,  less  in  the  one  case  than .  the  other ; 
and,  secondly,  that  the  weight  of  the  model  being  greater,  this  inertia 
bears  a  less  proportion  to  the  amount  of  work  required'  for  inclining  it 
than  in  the  other  case. 

The  effect  of  this  inertia  adding  itself  to  the  buoyancy  of  the  fluid, 
cannot  but  be  to  lift  the  vessel  out  of  the  water ;  and  to  cause  the.  -displace- 
ment to  be  less  at  the  termination  of  each  rolling  oscillation  than  at  its 
commencement.*  This  variation  in  volume  of  the  displacement  .was  appa- 
rent in  all  the  experiments.  Its  amount  was  measured  and  is  recorded 
in  the  last  column  of  the  Table ;  its  tendency  is  to  produce  in  the  body 
vertical  oscillations,  which  are  so  far  independent  of  its  rolling  motion 
that  they  will  not  probably  synchronise  with  it.  The  body,  displacing, 
when  rolling,  less  fluid  than  it  would  at  rest,  the  effect  of  the  weight* 
used  in  the  experiments  to  incline  it  is  thereby  increased,  and  thus  is 
explained  the  fact  (apparent  in  the  eighth  and  ninth  columns  of  the  Table) ; 
that  the  inclination  by  experiment  is  somewhat  greater  than  the  formula 
would  make  it. 

The  dynamical  stability  of  a  vessel  whose  athwart  sections  (where  they 


*  This  result  connects  itself  with  the  well-known  fact  of  the  rise  of  a  vessel 
out  of  the  water  when  propelled  rapidly,  which  is  so  great  in  the  case  of  fast 
track-boats,  as  considerably  to  reduce  the  resistance  upon  them. 

41 


APPENDIX. 


are  subject  to  immersion  and  emersion)  are  circular, 
a  common  axis. 

Fig.l. 


in 


Fig.  2. 


Let  EDF,  fig.  1.  or  2.,  be  an  athwart  section  of  such  a  vessel,  the 
parts  of  whose  periphery  ES  and  FR,  subject  to  immersion  and  emersion, 
are  parts  of  the  same  circular  arc  ETF,  whose  centre  is  0.  Let  G,  repre- 
sent the  projection  of  the  centre  of  gravity  of  the  vessel  on  this  section, 
and  G2  that  of  the  centre  of  gravity  of  the  space  whose  section  is  SDRT, 
supposing  it  filled  with  water.  The  space  lies  wholly  within  the  vessel  in 
fig.  1.  and  without  it  in  fig.  2.  Let 

A,  =  CG,,  Jit  =  CG,. 

W,  =  weight  of  vessel. 

W2  =  weight  of  water  occupying,  or  which  would  occupy,  the  space 
whose  section  is  STRD. 

6  =  the  inclination  from  the  vertical. 

Since  in  the  act  of  the  inclination  of  the  vessel  the  whole  volume  of 
the  displaced  fluid  remains  constant,  and  also  that  volume  of  which  STRD 
is  the  section,*  it  follows  that  the  volume  of  that  portion  of  which  the 
circular  area  PSRQ  is  the  section  remains  also  constant,  and  that  the 
water-line  PQ,  which  is  the  chord  of  that  area,  remains  at  the  same  dis- 
tance from  0,  so  that  the  point  0  neither  ascends  nor  descends.  Now  the 
forces  which  constituted  the  equilibrium  of  the  vessel  in  its  vertical  posi- 
tion were  its  weight  and  that  of  the  fluid  it  displaced.  Since  the  point  0 
is  not  vertically  displaced,  the  work  of  the  former  force,  as  the  body 
inclines  through  the  angle  0,  is  represented  by  —  Wj  A,  vers.  8.  The  work 
of  the  latter  is  equal  to  that  of  the  upward  pressure  of  the  water  which 
would  occupy  the  space  of  which  the  circular  area  PTQ  is  the  section 
increased,  in  the  case  represented  in  fig.  1.,  by  that  of  the  water  which 
would  occupy  STRD;  and  diminished  by  it  in  the  case  represented  in 
fig.  2. 

But  since  the  space,  of  which  the  circular  area  PTQ  is  the  section, 
remains  similar  and  equal  to  itself,  its  centre  of  gravity  remains  always 
at  the  same  distance  from  the  centre  0,  and  therefore  neither  ascends 
nor  descends.  "Whence  it  follows  that  the  work  of  the  water  which 
would  occupy  this  space  is  zero ;  so  that  the  work  of  the  whole  displaced 
fluid  is  equal  to  that-  of  the  part  of  it  which  occupies  the  space  STRD, 


*  It  will  be  observed  that  the  space  STRD  is  supposed  always  to  be  under 
water. 


ON    THE    ROLLING    OF    SHIPS.  -.  6-t3 

taken  in  the  case  represented  in  fig.  1 .  with  the  positive,  and  in  that  re- 
presented in  fig.  2.  with  the  negative  sign.  It  is  represented,  therefore, 
generally  by  the  formula  ±  W,  A2  vers.  0.  On  the  whole,  therefore,  the 
work  2u2  of  those  forces,  which  in  the  vertical  position  of  the  body  con- 
stituted its  equilibrium,  is  represented  by  the  formula — 
SUz  =  —  W,  A,  vers.  6  ±  W2  A2  vers.  8. 

Representing,  therefore,  the  dynamical  stability  2^  by  U  (0),  we  have  by 
equation  (2.  p.  122.) 

U  (9)  =  (W,  A,  T  W2  &2)  vers.  9, 

in  which  expression  the  sign  ~-f  is  to  be  taken  according  as  the  circular 
area  ATB  lies  wholly  within  the  area  ADB,  as  in  fig.  1  ,  or  partially  with- 
out it,  as  in  fig.  2.  Other  things  being  the  same,  the  latter  is  therefore  a 
more  stable  form  than  the  other. 

13.  The  work  of  the  upward  pressure  of  the  water  upon  the  vessel 
represented  in  fig.  2.  being  a  negative  quantity,  — W2AZ  vers.  0,  it  follows 
that  the  point  of  application  of  the  pressure  must  be  moved  in  a  direction 
opposite  to  that  in  which  the  pressure  acts ;  but  the  pressure  acts  upwards, 
therefore  its  point  of  application,  i.  e.  the  centre  of  gravity  of  the  displaced 
fluid,  descends.  This  property  may  be  considered  to  distinguish  mechani- 
cally the  class  of  vessels  whose  type  is  fig.  1.,  from  that  class  whose  type  is 
fig.  2. ;  as  the  property  of  including  wholly  or  only  partly,  within  the  area 
of  any  of  their  athwart  sections,  the  corresponding  circular  area  ETF,  dis- 
tinguishes them  geometrically. 

The  dynamical  stability  of  a  vessel  of  any  given  form  subjected  to  a  roll- 
ing or  pitching  motion. 

Conceive  the  vessel,  after  having  completed  an  oscillation  in  any  given 
direction — being  then  about  to  return  towards  its  vertical  position — to 
be  for  an  instant  at  rest,  and  let  RS  represent  the 
intersection  of  its  plane  of  flotation  then,  and  PQ 
of  its  flotation  when  in  its  vertical  position,  with 
a  section  CAD  of  the  vessel  perpendicular  to  the 
mutual  intersection  0  of  these  planes.     The  sec- 
tion CAD  will  then  be  a  vertical  section  of  the 
vessel. 

Let  G  be  the  projection  upon  it  of  the  vessel's 
centre  of  gravity  when  in  its  vertical  position. 

H,  that  of  the  centre  of  gravity  of  the  fluid  displaced  by  the  vessel  in  the 
vertical  position. 

g,  that  of  the  fluid  displaced  by  the  portion  of  the  vessel  of  which  QOS 
is  a  section.  * 

h,  that  of  the  fluid  which  would  be  displaced  by  the  portion,  of  which 
FOR  is  a  section,  if  it  were  immersed. 

GM,  HN,  gm,  Jin,  KL,  perpendiculars  upon  the  plane  RS. 

W  =  weight  of  vessel  or  of  displaced  fluid. 

to  =  weight  of  water  displaced  by  either  of  the  equal  portions  of  the 
vessel  of  which  FOR  and  QOS  are  sections. 


APPENDIX. 

H!  =  depth  of  centre  of  gravity  of  vessel  in  vertical  position. 

H,  =  depth  of  centre  of  gravity  of  displaced  water  in  vertical 

position. 

Hj  =  elevation  of  centre  of  gravity  of  vessel. 
Hj  =  elevation  of  centre  of  gravity  of  displaced  water. 
P  =  area  of  plane  PQ. 
0  =  inclination  of  planes  PQ  and  RS. 
*l  =  inclination  of  line  0  in  which  planes  PQ  and  RS  intersect, 

to  that  line  about  which  the  plane  PQ  is  symmetrical. 
h  =  perpendicular  distance  of  line  O  from  centre  of  gravity  of 

plane  PQ. 
£  =  inclination  to  horizon  of  line  about  which  the  plane  PQ  is 

symmetrical. 

x  =  distance  of  section  CAD,  measured  along  the  line  whose 
projection  is  O,  from  the  point  where  that  line  intersects 
the  midship  section. 


y,  =  RS. 
z  =  Tin  +  mg. 
jt  =  KL. 

I  =  moment  of  inertia  of  plane  PQ  about  axis  0. 

A  and  B  =  moments  of  inertia  of  PQ  about  its  principal  axes. 
p,  =  weight  of  a  cubic  unit  of  water. 

Suppose  the  water  actually  displaced  by  the  vessel  to  be,  on  the  contrary, 
contained  by  it;  and  conceive  that  which  occupies  the  space  QOS  to  pass 
into  the  space  POR,  the  whole  becoming  solid.  Let  AH3  represent  the 
corresponding  elevation  of  the  centre  of  gravity  of  the  whole  contained 
fluid.  Then  will  AH2  +  AH3  represent  the  total  elevation  of  the  centre  of 
gravity  of  this  fluid  as  it  passes  from  the  position  it  occupied  when  the 
vessel  was  vertical  into  the  position  PAQ.  But  this  elevation  is  obviously 
the  same  as  though  the  fluid  had  assumed  the  solid  state  in  the  vertical 
position  of  the  body,  and  the  latter  had  revolved  with  it,  in  that  state,  into 
its  present  position.  It  is  therefore  represented  by  KH  —  NH  ; 

/.  AH2  +  AH3  =  KH  —  NH  and  AH3  =  KH  —  NH  —  AH2. 

Since,  moreover,  by  the  elevation  of  the  fluid  in  QOS,  whose  weight  is  «o, 
into  the  space  OPR,  and  of  its  centre  of  gravity  through  (gm  +  Tin\  the 
centre  of  gravity  of  mass  of  fluid  of  which  it  forms  a  part,  and  whose  weight 
is  W,  is  raised  through  the  space  AH3  ;  it  follows,  by  a  well-known  property 
of  the  centre  of  gravity  of  a  system,*  *that 

*  The  line  joining  the  centres  of  gravity  of  the  vessel  and  its  immersed  part, 
in  its  vertical  position,  is  parallel  to  the  plane  CAD,  for  it  is  perpendicular  to 
the  plane  PQ,  to  vhose  intersection  with  the  plane  RS  the  plane  CAD  is  per- 
pendicular ;  .  •  .  GK  =  H,  and  HK  =  H* 


ON   THE   EOLLING    OF    SHIPS.  645 

W.  A  H3  =  w  (gm  +  hri)  • 
.-.  W(KH — NH  ~  A  H,)  =  w  (gm  +  hri). 
Bnt 

NH  =  KHcos.0— KL  =  H2  cos.  0—21 ; 

.-.  KH — N"H  =  H2  vers.  9  +  *, 
and 


.-.  W  (H2  vers.  0  +  ?i  —  A  H2)  =  wz ; 

.-.  W.  A  H,  =  W  (H2  vers.  0  +x)  —  wz  .  .  . .  (28). 
Also  A  H!  =  KG  —  MG  =  H,  —  (H,  cos.  0  —  ji)  =  H,  vers.  0 + x ; 

/.  W  (A  H,  —  A  H2)  =  W  (H,  —  H2)  vers.  0  +wz ; 

/.  (equation  2T.)  U  (0, »?)  =  W  (H,  —  Ha)  vers.  0  +«»;...  (29). 
If  aj3  be  a  vertical  prismatic  element  of  the  space  QOS,  whose  base 
is  dx  dy  cos.  0,  and  height  y  sin.  0  then  will  w  .mg  be  represented,  in 

respect  to  that  element,  by  p,y  sin.  0.  c£»  c?y  cos.  0-%y  sin.  0,  or  by  ^ ju sin.a0 

cos.  6  y*dx  dy ;  and  wz  will  be  represented,  in  respect  to  the  whole  space 
of  which  PrsQ  is  the  section,  by 

2  p  sin.8  e  cos.  Qjjy^  dy, 

or  by  2^  sm-a  0  cos- e- 1- 

If  therefore  we  represent  by  $  the  value  of  wz,  in  respect  to  the  spaces 
of  which  the  mixtilinear  areas  PRr  and  QSs  are  the  sections,  we  have 

wz  =  ^fil  sin.*  e  cos.  e + <j,. 

But  the  axis  O,  about  which  the  moment  of  inertia  of  the  plane  PQ  is 
I,  is  inclined  to  the  principal  axes  of  that  plane  at  the  angles  y  and  ^ — tj, 

about  which  principal  axes  the  moments  of  inertia  are  A  and  B, 
/.  I  =  A  cos.2  n  +  B  sin.2  tj  +  PA2, 


W  (Ht  —  H2)  vers.  0  +  '2>  (A  cos.2  ^  +  B  sin. 2 »;  +  PA2)  sin2  9  cos.  0 + 9 ...  (30). 
It  has  been  shown  by  M.  DUPIN*  that  when  6  is  small  the  line  in 

*  Sur  la  Stabilite  des  Corps  Flottarits,  p.  32.  In  calculations  having  refer- 
ence to  the  stability  of  ships,  it  is  not  allowable  to  consider  0  extremely  small, 
except  in  so  far  as  they  have  reference  to  the  form  of  the  ship  immediately 
about  the  load-water  line.  The  rolling  of  the  ship  often  extends  to  20°  or  30°, 
and  is  therefore  largely  influenced  by  the  form  of  the  vessel  beyond  these 
limits.  Generally,  therefore,  equation  30.  is  to  be  taken  as  that  applicable  to 
the  rolling  of  ships,  those  which  follow  being  approximations  only  applicable 
to  small  oscillations,  and  not  sufficiently  near  (excepting  equation  37)  for 
practical  purposes 


APPENDIX. 

which  the  planes  PQ  or  RS  intersect  passes  through  the  centre  of  gravity 

of  each  ;  in  this  case 

/.  I  =  A  cos.2  q  +  B  sin.2  q  ; 

therefore  by  equation  (30), 


If  e  be  so  small  that  the  spaces  PrR  and  QsS  are  evanescent  in  compari- 
son with  POr  and  QO,  then,  assuming  <j>  =  0  and  cos.  0  =  1, 

U  (0,  rj)  =  W  (H,  —  H2)  vers.  e  +  -  p  (A  cos.2  n  +  B  sin.2  n}  sin.2  0,  ...  (3  1), 
which  may  be  put  under  the  form 

U  (0,  if  )  =  j  W  (H,  —  H2)  +  JK  (  A  cos.2  n  +  B  sin.2  *?)  j  vers.  0. 
Again,  since 


sin.  £  =  sin.  flsin.j?,  ....  (32), 
and  (A  cos.2  ^  +  B  sin.2  17)  sin.2  0  =  {A  +  (B  —  A)sin.8»;}sm.20, 

.-.  (A  cos.2  n  +  B  sin.'jf)  sin.2  0  =  A  sin.2  e  -f  (B  -  A)  sin.a  C  ; 
/.by  equation  31, 

—  H2)vers.0-f  >{Asin.20  +  (B—  A)sin.2^},  ----  (33), 


ty  which  formula  the  dynamical  stability  of  the  ship  is  represented,  loth 
as  it  regards  a  pitching  and  a  rolling  motion. 

If  in  equation  31.  y  =  -,  the  line  in  which  the  plane  PQ  (parallel  to  the 
2 

deck  of  the  ship)  intersects  its  plane  of  flotation  is  at  right  angles  to  the 
length  of  the  ship,  and  we  have,  since  in  this  case  0  =  £  (see  equation  32.), 

U(0  =  W(H,  —  HO  vers£+  I^Bsin^  .....  (34), 

which  expression  represents  the  dynamical  stability,  in  regard  to  a  pitch- 
ing motion  alone,  as  the  equation 

U(0)  =  W(H,—  H2)  vers  0+  ~p  Asin20  .....  (35), 

represents  it  in  regard  to  a  rolling  motion  alone. 

16.  If  a  given  quantity  of  work  represented  by  TJ(«)  be  supposed  to 
be  done  upon  the  vessel,  the  angle  0  through  which  it  is  thus  made  to 

roll  may  be  determined  by  solving  equation  35.  with  respect  to  sin.-. 
We  thus  obtain 


2—  2^  A.U(0).  .  .  (36). 


ON   THE    ROLLING    OF    SHIPS.  617 

17.  If  PR  and  QS  be  conceived  to  be  straight  lines,  so  that  POR  and 
QOS  are  triangles,  then  w.  z,  taken  in  respect  to  an  element  included 
between  the  section  CAD,  and  another  parallel  to  it  and  distant  by  the 
small  space  dx,  is  represented  by 


or,  since  mg  +  nh=-  yl  sin.0, 

3 

by  _ 

12 

.-.wz  =  —p  sin.20  I  y] 

12  J 

and,  equation  29 

U(0,£)=  W(Hj—  H2)  vers.o  +  —  n&v.?e  fy]y$x,  .  .  .  (37), 
24          v 

which  formula  may  be  considered  an  approximate  measure  of  the  stability 
of  the  vessel  under  all  circumstances. 

If,  as  in  the  case  of  the  experiments  of  Messrs.  FINCHAM  and  RAWSON, 
the  vessel  b|  prismatic  and  the  direction  of  the  disturbance  perpendicular 
to  its  axis, 

y  =  constant  =  a,  and  z  =  —a  sin.fi  ; 
3 

.•.wz  =  —a/w  sin.  0,  and 
3 

!ow>  sinA 
3         • 

A  rigid  surface  on  which  the  vessel  may  ~be  supposed  to  rest  whilst  in  the 
act  of  rolling. 

If  we  imagine  the  position  of  the  centre  of  gravity  of  a  vessel  afloat 
to  be  continually  changed  by  altering  the  positions  of  some  of  its  con- 
tained weights  without  altering  the  weight  of  the  whole,  so  as  to  cause 
the  vessel  to  incline  into  an  infinite  number  of  different  positions  dis- 
placing, in  each,  the  same  volume  of  water,  then  will  the  different  planes 
of  flotation,  corresponding  to  these  different  positions,  envelope  a  curve  1 
surface,  called  the  surface  of  the  planes  of  flotation  (surface  des  flotaisons), 
whose  properties  have  been  discussed  at  length  by  M.  DUPIN  in  his  ex- 
cellent memoir,  Sur  la  Stabilite  des  Corps  Flottants,  which  forms  part  of 
his  Applications  de  Geometric.*  So  far  as  the  properties  of  this  surface 
concern  the  conditions  of  the  vessel's  equilibrium,  they  have  been  ex- 
hausted in  that  memoir,  but  the  following  property,  which  has  reference 


*  BACHBLIEB,  Paris,  1822. 


648 

rather  to  the  conditions  of  its  dynamical  stability  than  its  equilibrium,  is 
not  stated  by  M.  DUPIN  : — 

If  we  conceive  the  surface  of  the  planes  of  flotation  to  become  a  rigid 
surface,  and  also  the  surface  of  the  fluid  to  become  a  rigid  plane  without 
friction,  so  that  the  former  surface  may  rest  upon  the  latter  and  roll  and 
slide  upon  it,  the  other  parts  of  the  vessel  being  imagined  to  be  so  far  im- 
material as  not  to  interfere  with  this  motion,  but  not  so  as  to  take  away 
their  weight  or  to  interfere  with  the  application  of  the  upward  pressure  of 
the  fluid  to  them,  then  will  the  motion  of  the  vessel,  when  resting  by  this 
curved  surface  upon  this  rigid  but  perfectly  smooth  horizontal  plane,  be 
the  same  as  it  was  when,  acted  upon  by  the  same  force,  it  rolled  and  pitched 
in  the  fluid. 

In  this  general  case  of  the  motion  of  a  body  resting  by  a  curved  sur- 
face upon  a  horizontal  plane,  that  motion  may  be,  and  generally  will  be, 
of  a  complicated  character,  including  a  sliding  motion  upon  the  plane, 
and  simultaneous  motions  round  two  axes  passing  through  the  point  of 
contact  of  the  surface  with  the  planes  and  corresponding  with  the  rolling 
and  pitching  motion  of  a  ship.  It  being  however  possible  to  determine 
these  motions  by  the  known  laws  of  dynamics,  when  the  form  of  the 
surface  of  the  planes  of  flotation  is  known,  the  complete  solution  of  the 
question  is  involved  in  the  determination  of  the  latter  surface. 

The  following  property*,  proved  by  M.  DUPIN  in  the  memoir  before 
referred  to  (p.  32),  effects  this  determination : — 

"  The  intersection  of  any  two  planes  of  flotation,  infinitely  near  to  each 
other,  passes  through  the  centre  of  gravity  of  the  area  intercepted  upon 
either  of  these  planes  by  the  external  surface  of  the  vessel." 

If,  therefore,  any  plane  of  flotation  be  taken,  and  the  centre  of  gravity 
of  the  area  here  spoken  of  be  determined  with  reference  to  that  plane  of 
flotation,  then  that  point  will  be  one  in  the  curved  surface  in  question, 
called  the  surface  of  the  planes  of  flotation,  and  by  this  means  any  number 
of  such  points  may  be  found  and  the  surface  determined. 

The  axis  about  which  a  vessel  rolls  may  be  determined,  the  direction  in 
which  it  is  rolling  being  given. 

If,  after  the  vessel  has  been  inclined  through  any  angle,  it  be  left  to 
itself,  the  only  forces  acting  upon  it  (the  inertia  of  the  fluid  being  neglected) 
will  be  its  weight  and  the  upward  pressure  of  the  fluid  it  displaces ;  the 
motion  of  its  centre  of  gravity  will  therefore,  by  a  well-known  principle 
of  mechanics,  be  wholly  in  the  same  vertical  line. 

«  Let  HE  represent  this  vertical  line,  PQ  the  surface  of  the  fluid,  and 
aMb  the  surface  of  the  planes  of  flotation.  As  the  centre  of  gravity  G 
traverses  the  vertical  HK,  this  surface  will  partly  roll  and  partly  slide 
by  its  point  of  contact  M  on  the  plane  PQ. 

If  we  suppose,  therefore,  PRQ  to  be  a  section  of  the  vessel  through 

*  This  property  appears  to  have  been  first  given  by  EULBR. 


ON   THE   ROLLING   OF   SHIPS. 


619 


the  point  M,  and  perpendicular  to  the  axis  about  which  it  is  rolling,  and 
if  we  draw  a  vertical  line  MO  through  the  point  M,  and  through  G  a 
horizontal  line  GO  parallel  to  the  plane  PRQ,  then 
the  position  of  the  axis  will  be  determined  by  a  line 
perpendicular  to  these,  whose  projection  on  the  plane 
PRQ  is  O. 

For  since  the  motion  of  the  point  G  is  in  the  verti- 
cal line  HK,  the  axis  about  which  the  body  is  revolv- 
ing passes  through  GO,  which  is  perpendicular  to 
HK ;  and  since  the  point  M  of  the  vessel  traverses 
the  line  PQ,  the  axis  passes  also  through  MO,  which 
is  perpendicular  to  PQ ;  and  GO  is  drawn  parallel  to,  and  MO  in  the 
plane  PRQ,  which,  by  supposition,  is  perpendicular  to  the  axis,  therefore 
the  axis  is  perpendicular  to  GO  and  MO. 

If  HK  be  in  the  plane  PRQ,  winch  is  the  case  whenever  the  motion  is 
exclusively  one  of  rolling  or  one  of  pitching,  the  point  0  is  determined  by 
the  intersection  of  GO  and  MO. 

The  time  of  the  rolling  through  a  small  angle  of  a  vessel  whose  athwart 
sections  are  (in  respect  to  the  parts  subject  to  immersion  and  emersion) 
circular,  and  have  their  centres  in  the  same  longitudinal  axis. 

Let  EDF  (fig.  1.  or  fig.  2J>  represent  the  midship  section  of  such  a 


vessel,  in  which  section  let  the  centre  of  gravity  GI  be  supposed  to  be  situ- 
ated, and  let  HK  be  the  vertical  line  traversed  by  G,  as  the  vessel  rolls. 
Imagine  it  to  have  been  inclined  from  its  vertical  position  through  a  given 
angle  QI  and  the  forces  which  so  inclined  it  then  to  have  ceased  to  act 
upon  it,  so  as  to  have  allowed  it  to  roll  freely  back  again  towards  its  posi- 
tion of  equilibrium  until  it  had  attained  the  inclination  OOD  to  the  verti- 
cal, which  suppose  to  be  represented  by  0. 

Referring  to  equation  1.  page  123.  let  it  be  observed  that  in  this  case 
2w2=0,  so  that  the  motion  is  determined  by  the  condition 

Zu^  —  Zwv* (38). 


But  the  forces  which  have  displaced  it  from  the  position  in  which  it 
was,  for  an  instant,  at  rest  are  its  weight  and  the  upward  pressure  of  the 


650  APPENDIX. 

water;  and  the  work  of  these,  U(0,)  —  U(0),  done  between  the  inclinations 
6  and  0,  when  the  vessel  was  in  the  act  of  receding  from  the  vertical,  was 
shown  to  be  represented  by  (W  A  =F  Wz^j)  (vers.  0  —  vers.  0,)  ;  therefore 
the  work,  between  the  same  inclinations,  when  the  motion  is  in  the 
opposite  direction,  is  represented  by  the  same  expression  with  the  sign 
changed  ; 

(vers.  0,  —  vers.  0), 


and  since  the  axis  about  which  the  vessel  is  revolving  is  perpendicular  to 
the  plane  EDF,  and  passes  through  the  point  O,  if  W,&2  represents  its 
moment  of  inertia  about  an  axis  perpendicular  to  the  plane  EDF,  and 
passing  through  its  centre  of  gravity  G1} 


Substituting  in  equation  38.  and  writing  for  OG,  its  value  A,  sin.  <?,  we 
have 


vers.  0,  —  vers.  fl)=  -1  (*»+A»mn.«0)  (—  )  ; 


.  .  .  (39). 


or  assuming  0  to  be  so  small  that  the  fourth  and  all  higher  powers  ol 
sin.  —  0  may  be  neglected,  and  observing  that,  this  being  the  case, 


y 


*«  sec.2 


. 


ON   THE   ROLLING   OF   SHIPS.  651 


~lc\J  1  +  ~V~sin-V-' 


W  —  — r  tfnr^     I     ' — /  d  sin.  —  i 


.8-^—  sin.a- 


and 


4 

/ 


The  sign  +  being  taken  according  as  the  centre  of  gravity  of  the  displaced 
fluid  ascends  or  descends. 

The  time  of  a  vessel's  rolling  or  pitching  through  a  small  angle,  its  form 
and  dimensions  ~being  any  whatever. 

Let  EDF  (figs.  1.  or  2.)  represent  the  midship  section  of  such  a  vessel, 
supposed  to  be  rolling  about  an  axis  whose  projection  is  0  ;  and  let  0 
represent  the  centre  of  the  circle  of  curvature  of  the  surface  of  its  planes 
of  flotation  at  the  point  M  where  that  surface  is  touched  by  the  plane  PQ, 
being  above  the  load  water-line  AB  in  fig.  1,  and  beneath  it  in  fig.  2.  Let 
the  radius  of  curvature  CM  be  represented  by  p  ;  then  adopting  the  same 
notation  as  in  the  last  article,  and  observing  that  the  axis  O  about  which 
the  vessel  is  turning  is  perpendicular  to  EDF,  we  shall  find  its  moment  of 
inertia  to  be  represented  by 


where  H,  represents  the  depth  of  the  centre  of  gravity  in  the  vertical  posi- 
tion of  the  vessel. 


652  APPENDIX. 

Also,  by  equation  35. 
^i=U0i)__U(0)=WI(H1—  H2)(cos.0—  cos.eO  +  - 


.•.by  equation  38. 


IP  +  (H,— p)2  sin.2  0 


(H,-H2)  (cos.  0-cos.  et) +:  cos.8  0  —  cos.1*,) 


— p)a8iD.'Q 


•j  H!— H2  +  ^%  (COS.  0  +  COS.  0,)  U  COS.  0  —  COS.  0,  j. 


Assuming  0  and  9l  to  be  so  small  that  cos.  0  +  cos.  el  =  2,  and  observing 
that 


cos>  e  —  cos.  0j  =  vers.  0!  —  vers.  0, 


Supposing,  moreover,  p  to  remain  constant  between  the  limits— 0,  and 
and  integrating  as  in  equation  39. 


iD,,      .  .  .  (41). 


Since  the  value  of  sin.*  ~  Ql  is  exceedingly  small,  the  oscillations  are 

A    . 

nearly  tautochronous,  and  the  period  of  each  is  nearly  represented  by  the 
formula 


....  (42.) 


EQUILIBRIUM   OF   PRESSURES.  653 

The  following  method  is  given  by  M.  DTJPIN  for  determining  the  value 
ofp*:— 

"  If  the  periphery  of  the  plane  of  flotation  be  imagined  to  be  loaded  at 
every  point  with  a  weight  represented  by  the  tangent  of  the  inclination  of 
the  sides  of  the  vessel  at  that  point  to  the  vertical,  then  will  the  moments 
of  inertia  of  that  curve,  so  loaded,  about  its  two  principal  axes,  when 
divided  by  the  area  of  the  plane  of  flotation,  represent  the  radii  of  greatest 
and  least  curvature  of  the  envelope  of  the  planes  of  flotation." 

If  p  be  taken  to  represent  the  radius  of  greatest  curvature,  the  formula 
41.  will  represent  the  time  of  the  vessel's  rolling;  if  the  radius  of  least 
curvature  (B  being  also  substituted  for  A),  it  will  represent  the  time  of 
pitching. 

NOTE  D. 

On  the  conditions  of  the  equilibrium  of  any  number  of  pressures  in  the 
same  plane,  applied  to  a  body  movedble  about  a  cylindrical  axis  in  the  state 
"bordering  upon  motion.  (From  a  memoir  on  the  Theory  of  Mechanics, 
printed  in  the  second  part  of  the  Transactions  of  the  Royal  Society  for  1841.) 

LET  PI,  P2,  P3,  &c.  represent  these  pressures,  and  R  their  resultant.  Also 
let  al:  a2,  «3,  represent  the  perpendiculars  let  fall  upon  them  severally  from 
the  centre  of  the  axis,  those  perpendiculars  being  token  with  the  positive 
signs  whose  corresponding  pressures  tend  to  turn  the  system  in  the  same 
direction  as  the  pressure  P,,  and  those  negatively  which  tend  to  turn  it  in 
the  opposite  direction.  Also  let  A,  represent  the  perpendicular  distance  of 
the  direction  of  the  resultant  R  from  the  centre  of  the  axis,  then,  since  R 
is  equal  and  opposite  to  the  resistance  of  the  axis,  and  that  this  resistance 
and  the  pressures  P,,  P2,  P3,  &c.  are  pressures  in  equilibrium,  we  have  by 
the  principle  of  the  equality  of  moments, 

PI«I  +  P2«2  +  P^3  +  &c.  =  *,R. 

Representing,  therefore,  the  inclinations  of  the  directions  of  the  pressures 
PI,  P2,  P3,  &c.  to  one  another  by  <r2,  «, .„  i23,  f,  &c.,  &c.,  and  substituting 
for  the  value  of  R.J 


*  Applications  de  Geometrie,  p.  47. 

f  The  inclination  i,.a  of  the  directions  of  any  two  pressures  in  the  above  ex- 
pression is  taken  on  the  supposition  that  both  the  pressures  act  from,  or  both 
towards  the  point  in  which  they  intersect,  and  not  one  towards,  and  the  other 
from,  that  point ;  so  that  in  the  case  represented  in  the  figure  in  the  note  at  p. 
171.,  the  inclination  »,.,  of  the  pressures  P,  and  P2,  represented  by  the  arrows, 
is  not  the  angle  P,  IP2,  but  the  angle  P,IQ,  since  IQ  and  IP,  are  directions  of 
these  pressures,  both  tending  from  this  point  of  intersection,  whilst  the  direc- 
tions of  P2T  and  IP,  are  one  of  them  towards  that  point,  and  the  other  from  it 

\  POISSON,  Mecanique,  Art.  33. 


APPENDIX. 


.p.= 


+  2  P,P2  cos.  t1<2  +  2  PjPs  cos.  i,.,  + 
+  2  P2P3  cos.  i2.3  +  2  P2P4  cos.  t2M  + 

+  &C.  &C. 

f  P,2  +  2  P!  (P2  COS.  »,.,  +  P3  COS. 


+  &C.  &C. 


If  the  value  of  P,  involved  in  this  equation  be  expanded  by  Lagrange's 
theorem  *,  in  a  series  ascending  by  powers  ofa,,  and  terms  involving  powers 
above  the  first  be  omitted,  we  shall  obtain  the  following  value  of  that 
quantity  :  — 


P  —  _ 


1    * 


a, 


or  reducing, 
P,= 


(P2COS.t,.2 


+  2  PaP3  cos.  i2.3  +  2  P2P4  cos.  <2.4 
+  2P3P4cos.«3.4  +  ---- 


.  tr2  +  a,,2) 


P22(«,2  — 


+  &c.  &c. 

+  2  P,  P3{02«3  —  «,(«,  cos.t2.3  +  «2  cos.  t,.8  +  «3  cos.  i, 

4-2PtP4{a»a4  — 

+  &C.  &C. 


Now  a7 — 2«,a2cos.  «,.2+«2  represents  the  square  of  the  line  joining  the 
feet  of  the  perpendiculars  <z,  and  «2  let  fall  from  the  centre  of  the  axia 
upon  Pj  and  P2;  similarly  a\ — 2a,a3  cos.  1,.,  +  ag  represents  the  square  of 
the  line  joining  the  feet  of  the  perpendiculars  let  fall  upon  P!  and  P3,  and 


*  This  expansion  may  be  effected  by  squaring  both  sides  of  the  equation, 
solving  the  quadratic  in  respect  to  P,,  neglecting  powers  of  >  above  the  first 
and  reducing ;  this  method,  however,  is  exceedingly  laborious. 


ROLLING   MOTION    OF   A   CYLINDER.  655 

BO  of  the  rest.    Let  these  lines  be  represented  by  LL2,  Lt  3,  L,.4,  &c.,  and  let 
the  different  values  of  the  function 


al  #,  cos. 
be  represented  by  M2.3,  M2.4,  M3.4,  &c. 


NOTE  E. 

ON  THE   KOLLING  MOTION  OF   A   CYLINDER. 

(From  a  memoir  printed  in  the  Transactions  of  the  Royal  Society  for 
1851,  part  II.) 

THE  oscillatory  motion  of  a  heterogeneous  cylinder  rolling  on  a  horizontal 
plane  has  been  investigated  by  EULER.*  He  has  determined  the  pressure 
of  the  cylinder  on  the  plane  at  any  period  of  the  oscillation,  and  the  time 
of  completing  an  oscillation  when  the  arcs  of  oscillation  are  small. 

The  forms  under  which  the  cylinder  enters  into  the  composition  of 
machinery  are  so  various,  and  its  uses  so  important,  that  I  have  thought  it 
desirable  to  extend  this  inquiry,  and  in  the  following  paper  I  have  sought 
to  include  in  the  discussion  the  case  of  the  continuous  rolling  of  the  cylin- 
der, and  to  determine— 

1st.  The  time  occupied  by  a  heterogeneous  cylinder  in  rolling  continu- 
ously through  any  given  space. 

2ndly.  The  time  occupied  in  its  oscillation  through  any  given  arc. 

Srdly.  Its  pressure,  when  thus  rolling  continuously,  on  the  horizontal 
plane  on  which  it  rolls. 

Under  the  second  and  third  heads  this  discussion  has  a  practical  appli- 
cation to  the  theory  of  the  pendulum ;  determining  the  time  occupied  in 
the  oscillations  of  a  pendulum  through  any  given  arc,  whether  it  rests 
on  a  cylindrical  axis  or  on  knife-edges,  and  the  circumstances  under 
which  it  will  jump  or  slip  on  its  bearings ;  and  under  the  first  and  third, 
to  the  stability  and  the  lateral  oscillations  of  locomotive  engines  in  rapid 
motion,  whose  driving-wheels  are,  by  reason  of  their  cranked  axles,  untruly 
balanced. 


*  Nova  Acta  Acad.  Petropol.  1788.     "  De  motu  oscillatorio  circa  axcm  cylin- 
dricum  piano  horizontal!  incumbentem." 


APPENDIX, 

Let  AMB  represent  the  section  of  a  heterogeneous  cylinder  through 
its  centre  of  gravity  G  and  perpendicular  to  its  axis  0 ;  and  let  M  be  its 
point  of  contact,  at  any  time,  with  the  hori- 
zontal plane  BD   on   which   it    is  rolling. 
Assume 

a  =  AC,  h  =  CG,  0  =  ACM. 
W  =  weight  of  cylinder.     W&*  =  momen- 
tum of  inertia  of  the  cylinder  ahout 
an    axis    passing   through   G  and 
parallel  to  the  axis  of  the  cylinder. 

w  =-  given  value  of  the  angular  velocity  (  —  )  when  e  has  the  given 

.  \  dt  / 

value  0j. 

y,  =  given  value  of  6  when  the  angular  velocity  has  the  given  value  «. 
I  =  given  value  of  GM  corresponding  to  the  value  0,  of  9. 


Then  W  (#*  +  GM2)  =  W(P  +  a?  +  h*  —  2ah  cos.  0)  =  moment  of  inertia 
ahout  M.  Since  moreover  the  cylinder  may  be  considered  to  be  in  the  act 
of  revolving  about  the  point  M  by  which  it  is  in  contact  with  the  plane, 
one-half  of  its  vis  viva  is  represented  by  the  formula 


and  one-half  of  the  vis  viva  acquired  by  it  in  rolling  through  the  angle 
0,  —  0,by 

_  2aA  cos.  0  +  #)*     — 


But  the  vertical  descent  of  the  centre  of  gravity  while  the  cylinder  is 
passing  from  the  one  position  into  the  other,  is  represented  by 

h  (cos.  0  —  cos.  0j). 
Therefore,  by  the  principle  of  vis  viva,* 

I  -j  (V  +  a9  —  2ah  cos.  e  +  F/  -)  —  (F  +  17)  w2  i  =  W£  (cos.  e  -  cos.  0,), 

whence  we  obtain 

(cos.  6  —  cos.  0Q  +  (ff  +  P)  co8 


&  V  _ 

dt/  ~ 


cos.  e  —  ( cos.  0,  —  ^-±1  w2 

fff\ V         *        2gh 

\aj    I  (ft      a      h\ 


*  POK60N,  Dynamique,  2me  partie,  565. ;  PONCELET,  Mecanique  Induatrielle, 
or  Art.  (129.)  of  this  Work. 


ROLLING    MOTION    OF    A    CYLINDER.  657 


^ 


—  .3 


0 

where  t  represents  the  time  of  the  body's  passing  from  the  inclination  0j  to 
zero. 

Now  let  it  be  observed  that  in  this  function  a>3  so  long  as  a  is  less  than 
g,  since 

#  +  ?>_(#»  +  p)M«?  or  ff  +  a*—  2ah  cos.  0,  +  #>  — 
and  .  •  .  #*  +  a2  +  A8>  2«A  cos.  0  —  (F  +  Z>8, 

a    h 
and 


1+a  1  —  a  a  —  COS. 


1_0  cos.0— p 

Then  when  0  =  0,      #2  sec.8  0  =  7 — -  =  ?*,    . ' .  sec.  0  =  1  and  0  =  0. 

1 — p 

When  9  =  0!  let  0  =  0,, 

af  Tfc4 

_  a  —  cos.  0i 


also 
.  __  l-q_ 

= 


a  —  cos.  0 
And  since — - 

t  2  cos.  0  — 


(a  —  j8)  ~cos.0  +  ? 

(a  +  3)(cos.8  ^  +  g2)  +  (a  —  J3)(C08.'  9—f) 
-.2008.0  =  —  (COS.2^^2) 


658  APPENDIX. 


2  .      (cos.2  y  +  g*)2-(<*  cos.2 


(cos.*  0  +  £*  —  a  cos.2 


Now 

^0  _     <Z0        <?  cos.  0    <?  cos.  ^  _  sin.  ^    <?  cos.  5 

^  ~~^  cos.  0  *  d  cos.  9  '      ^>      ~  sin.  0  "  ^  cos.  $  '  '  "  "  '  *' 

Also  by  equation  (6.), 


—  2(aC03.20  +j3^2)  COS.  ?>_2(a  -  13)^008. 


^  cos.  $~~  (g*  +  cos.2 

.  •  .  by  equations  (7.)  and  (8.), 
<?9  __  2(g  —  3)g2 

<^~  (1— 


2(g  —  J3)g  COS.  0 


/a  —  COS.  ')  y^_2fg  —  j3)^2      j  _  1  _  I 

'  '  V  cos.  0—  j3  )  Tt>~  (1  —  j3*)*"  '   (  (^4-cos.2  ?.)  (^2  +^>2  cos.8  0)j  j 

2(a  —  p)g2     j  _  1  _  | 
(1—  (32)J    '   j^  +  l-sin.^X^+p2—  ^2sin.2^)i  j" 


2(>-p)g'  j  _  1  _  ) 

~(1  —  3'Oi(^  +  ?2K1  +  ^)(  (1—  wsin.2^)(l—  c2sin.2^)i  f 


and 

1  +a 


ROLLING   MOTION    OF   A    CYLINDER.  950 

0i 

y'  /a  —  cos.  0\*'< 
(  — ~0~^T# )  " 

2(a  — /,_ 

n  sin.2  0)(1  —  c2  sin.2 
o 

2(a  — %2 


where  n(  —  nc^)  is  that  elliptic  function  of  the  third  order  whose  par* 
meter  is  — n  and  modulus  c. 

1 /T-^" 


#    a 


.  •  .  by  equations  11.  and  4. 


where  (9.)  (2.)  (3.) 

1  — cos.  0, 


„ 


*  I  cannot  find  that  this  function  has  before  been  integrated,  except  in  the 
case  in  which  0  is  exceedingly  small. 


APPENDIX, 
and  (10.)  (2.)  (3.) 

.j 


(a—/?) 


The  value  of  n(— TMJ^)  being  determinable  by  known  methods 
DBE,  Fonctions  Elliptiques,  vol.  i.  chap,  xxxiii.),  the  time  of  rolling  is  given 
by  equation  13. 

In  the  case  in  which  the  rolling  motion  is  not  continuous  but  oscillatory, 

we  have  w  =  0 ;  and  therefore  (equation  5.)  $,  =  -  ;  n  ( —  nc<p^)  becomes 

therefore  in  this  case  a  complete  function. 

To  express  the  value  of  this  complete  elliptic  function  of  the  third  order 
in  terms  of  functions  of  the  first  and  second  orders,  let 


Then* 


Eepresenting  therefore  the  time  of  a  semi-oscillation  by 

*!= 


where  (15.)  t  =    JvZn    vers-  »i  •  •  •  •  (18). 


Since  the  values  of  elliptic  functions  of  the  first  and  second  orders, 
having  given  amplitudes  and  moduli,  are  given  by  the  tables  of  LEGENDEE,- 
it  follows  that  the  value  of  t  is  given  by  this  formula  for  all  possible  values 
of  <j  and  4,. 

If  the  angle  of  oscillation  0,  be  very  small  e  is  very  small,  so  that  its 
square  may  be  neglected  in  comparison  with  unity.  In  this  case 

*  LBOBNDRE,  Calcus  des  Fonctions  Elliptiques,  vol.  i.  chap,  xxni  Art.  116. 


ROLLING   MOTION    OF   A   CYLINDER.  661 


=  Ee*  =  4,  and  Fc  ?  =  Ec^  =  £ 
2  22 


/.  Fe  -  £04,  —Ec  -  Fo^  =  0. 
2  2 


For  small  oscillations  therefore 


2 


If  the  pendulum  oscillate  on  knife-edges  a  =  0,  I  —  h,  and  we  obtain  the 
well-known  theorem  of  LEGENDEK  (Fonctions  Elliptiques,  vol.  i.  chap. 
viii.) 


where  (18.)  <?,  =  |  vers."  0,  =  sin.3  £i 


.  •  .  c  =  sin.  -  BI  .  .  .  .  (21). 

In  the  case  of  the  small  oscillations  of  a  pendulum  resting  on  knife-edge, 
equation  20.  becomes 


which  is  the  well-known  formula  applicable  to  that  case. 
If  the  pendulum  be  one  which  for  small  arcs  beats  seconds  (21.), 


.-.(20.)  2*  =  — — , (23). 


by  which  equation  the  time  of  the  oscillation  through  any  arc,  of  a  pen- 
dnluni  which  oscillates  through  a  small  arc  in  one  second,  may  be  deter- 
mined. I  have  caused  the  following  table  to  be  calculated  from  it. 


662 


APPENDIX. 


Table  of  the  time  occupied  in  oscillating  through  every  two  degrees  of  a 
complete  circle,  by  a  pendulum  which  oscillates  through  a  small  arc  in 
one  second. 


Arc  of  oscillation 
on  each  side  of 
the  vertical 
in  degrees. 

Time  of  one 
complete 
oscillation  in 
seconds. 

Arc  of  oscillation 
on  each  side  of 
the  vertical 
in  degrees. 

Time  of  one 
complete 
.    oscillation  in 
seconds. 

Arc  of  oscillation 
on  each  side  of 

in  degrees. 

Time  of  one 
complete 
oscillation  in 
seconds. 

2 

1-1899 

32 

1-3905 

62 

1-8032 

4 

1-2000 

34 

1-4089 

64 

1-8478 

6 

1-2123 

36 

1-4283 

66 

1-8963 

8 

1-2210 

38 

1  -4486 

68 

1-9491 

10 

1-2322 

40 

1-4698 

70 

2-0075 

12 

1  2439 

42 

1-4922 

72 

2-0724 

14 

1-2560 

44 

1-5157 

74 

2-1453 

16 

1-2686 

46 

1-5405 

76 

2-2285 

18 

1-2817 

48 

1-5667 

78 

2-3248 

20 

1-2953 

50 

1-5944 

80 

2-4393 

22 

1-3099 

52 

1-6238 

82 

2-5801 

24 

1-3249 

54 

1-6551 

84 

2-7621 

26 

1-3400 

56 

1-6884 

86 

3-0193 

28 

1-3560 

58 

1-7240 

88 

3-4600 

30 

1-3729 

60 

1-7622 

90' 

Infinite. 

The  pressure  of  the  cylinder  on  its  point  of  contact  with  the  plane  on  which 

it  rolls. 

Let  A'  be  the  point  where  the  point  *A  of  the  cylinder  was  in  contact 
with  the  plane. 

Let  A'N  =  x,  NQ  =  y. 

—  X  =  horizontal  pressures  on  M  in  direction 
A'M. 

Y  =  vertical  pressure  on  M  in  direction  MC. 

M  M  A'    D  Since  the  centre  of  gravity  G  moves  as  it  would 

do  if,  the  whole  mass  being  collected  there,  all  the  impressed  forces  were 
applied  to  it,  we  have,  by  the  principle  of  D'ALEMBEKT, 


dt* 


v 


....  (28). 


But  since  CA  =  a,  OG  =  h,  MOA  =  0, 
.  * .  x  =  ae  —  h  sin.  0, 
y  =  a  —  h  cos.  0 ; 
dx  dd 


Art.  96,  equations  (73.),  (74.). 


ROLLING   MOTION   OF   A   CYLINDER. 

dy     _   .      dd 


d'e 


>  .  . . .  (29). 


.  • .  by  equation  (29.), 


=  MA  cos.  0— ISta  sin.  0 ; 


by  equation  (28.), 


X=—  |  — 


— 


|  —  cos.eU 


(30). 


But  by  equation  (1.),  substituting  —  0  and  —  Bl  for  0  and 

QQB.gQ  +  (y+P)oa 
2a&  cos.  0  +  A2  '  *    ;* 


2afc(cos.  0  —  cos.  0,)  +  (7c2  +  V}  — 

g_ 

—  2ah  cos.  9  +  h2 


*® 


k2  +  a?  —  2ah  cos.  0  +  h2 


•'•       ~ 


..(32). 


Observing  that  a2  +  ft1  —  2ah  cos.  0,  =  I2. 
Differentiating  this  equation  and  dividing  Ibtfl  —\ 


—  Ism. 


_ 

(F  +  a7  +  7?-2aA  cos.  0)2 


C33X 


664  APPENDIX. 

Substituting  these  values  of  M  and  N  in  equation  (30.),  and  reducing, 

_.      WAsm.0(          (F  +  Z'X&'  +  A2—  ahcos.e)(g  +  au>T)> 
~t~~  *      *  " 

Whia 


'-  2oA  ooa. 


The  rotation  of  a  "body  about  a  cylindrical  axis  of  small  diameter. 

Assuming  a  =  0  in  equations  (31.),  (33.),  and  0!=0,  we  have 

2gft(cos.0—  1)      2  gr^  sin,  e 

2  =      >• 


Therefore,  by  equation  (30.), 
WAjgr^—  3cos.e) 

X=Y  \  —       — 


The  last  equation  may  be  placed  under  the  form 


If  --I       ,    w2  —  1  J  be  numerically  less  than  unity,  whether  it  be  positive 

or  negative,  there  will  be  some  value  of  0  between  0  and  n  for  which  this 
expression  will  be  equalled,  with  an  opposite  sign,  by  cos.  0,  and  for  which 
the  first  term  under  the  bracket  in  the  value  of  Y  will  vanish.  This  cor- 
responds to  a  minimum  value  of  Y  represented  by  the  formula 


But  if  -/  w2  —  1  J  be  numerically  greater  than  nnity,  then  the 

minimum  of  Y  will  be  attained  when  0  =  *,  and  when 


ROLLING   Ml/i'ION    OF   A   CYLINDEE.  G65 


The  Jump  of  an  Axis. 

If  Y  be  negative  in  any  position  of  the  body,  the  axis  will  obviously 
jump  from  its  bearings,  unless  it  be  retained  by  some  mechanical  expe- 
dient not  taken  account  of  in  this  calculation.  But  if  Y  be  negative  in 
any  position,  it  must  be  negative  in  that  in  which  its  value  is  a  minimum. 
If  a  jump  take  place  at  all,  therefore,  it  will  take  place  when  Y  is  a  mini- 
mum; and  whether  it  will  take  place  or  not,  is  determined  by  finding 
whether  the  minimum  value  of  Y  is  negative.  If  therefore  the  expression 
(42.)  or  (43.)  be  negative,  the  axis  will  jump  in  the  corresponding  case. 
An  axis  of  infinitely  small  diameter,  such  as  we  have  here  supposed, 
becomes  a  fixed  axis;  and  the  pressure  upon  a  fixed  axis,  supposed  to 
turn  in  cylindrical  bearings  without  friction,  is  the  same,  whatever  may 
be  its  diameter;  equations  (40.)  and  (41.)  determine  therefore  that  pres- 
sure, and  equation  (42.)  or  (43.)  determines  the  vertical  strain  upon  the 
collar  when  the  tendency  of  the  axis  to  jump  from  its  bearings  is  the 
greatest. 


The  Jump  of  a  Rolling  Cylinder. 

Whether  a  jump  will  or  will  not  take  place,  has  been  shown  to  be  deter- 
mined by  finding  whether  the  minimum  value  of  Y  be  negative  or  not. 

Substituting  a  for-/ — +  -  + f  |  and  reducing,  equation  (35.)  becomes 
Z\ah    a    hi 

y-W/1     *>  WF  +     C+a*8      cos.2e-2acos.0  +  l 

Wl- 


.-.-^=0,     Is  ^  ^  ^ 

Srdly,  when  0  =  0. 
The  first  condition  evidently  yields  a  positive  value  of  -r^-,  since  it 


666  APPENDIX. 

causes  the  first  term  of  the  preceding  equation  to  vanish  ;  and  the  second 
term  is  essentially  positive,  a  being  always  greater  than  unity. 

If,  therefore,  the  first  condition  be  possible,  or  if  there  be  any  value  of 
6  which  satisfies  it,  that  value  corresponds  to  a  position  of  minimum  pres- 
sure. Solving,  in  respect  to  cos.  0,  we  obtain 


The  first  condition  will  therefore  yield  a  position  of  minimum  pres- 
sure, if 


o  — i  / «~rrz ^   .  i   or  if 

or  if 


or  if 


J*\  („       iv  ^  (i*  _L  7-^  ^        il —  ^  • 

t^a  — 1J  ^AJ   +  I )  (a, —  I)      a 

and 


or  «• 


whence,  substituting  for  a  and  reducing,  we  obtain  finally,  the  conditions' 
\      {&2  +  (a  +  A)T          Sg\  (g\       {#+(»  —  &)'}'          fQ\ 

k^m^^-™)™^^ 

Of  these  inequalities  the  second  always  obtains,  because 


whatever  be  the  values  of  &,  a  and  A.    And  the  first  is  always  possible, 
since 

{F  +  (a  +  £)<}*  >(&*  +  ?*)  }#  +  (<,  _&)«}. 

^  If  the/rsi  obtain,  there  are  two  corresponding  positions  of  OA  on  either 
•ide  of  the  vertical,  determined  by  equation  (46.),  in  which  the  pressure  Y 
of  the  cylinder  upon  the  plane  is  a  minimum. 


ROLLING   MOTION   OF   A   CYLINDER.  667 

Substituting  the  other  two  values  (rt  and  0)  of  6  which  cause  -j-  to 


vanish  in  the  value  of  —^-^  we  obtain  the  values 

w 

<  h 


a  2ga\*  +  1)2  la  2gd\a—  I)2 

or 


_  a—  .  oa  „« 

2  I)2         $'* 


which  expressions  are  both  negative  if  the  inequalities  (47.)  obtain.  The 
same  conditions  which  yield  minimum  values  of  Y  in  two  corresponding 
oblique  positions  of  CA,  yield,  therefore,  maximum  values  in  the  two  ver- 
tical positions  ;  so  that  if  the  inequalities  (48.)  obtain,  there  are  two  posi- 
tions of  maximum  and  two  of  minimum  pressure. 

Substituting  the  values  of  cos.  0  (equation  46.)  in  equation  (44.),  and 
reducing,  we  obtain  for  the  minimum  value  of  Y  in  the  case  in  which  the 
inequalities  (48.)  obtain, 


+3 

If  this  expression  be  negative  the  cylinder  will  jump. 
In  the  case  in  which  «  =  0,  which  is  that  of  a,  pendulum  having  a  cylin- 
drical axis  of  finite  diameter,  it  becomes 


Y—  — 5 

If  the  first  of  the  inequalities  (48.)  do  not  obtain,  no  position  of  mini- 
mum pressure  corresponds  to  equation  (46.)  ;  and  the  inequalities  (47.)  do 

not  obtain,  so  that  the  values  (49.)  of  -=-5-,  given  respectively  by  the  sub- 
stitution of  rt  and  0  for  0,  are  no  longer  both  negative,  but  the  second  only. 
In  this  case  the  value  rt  of  0  is  that,  therefore,  which  corresponds  to  a  posi- 
tion of  minimum  pressure,  which  minimum  pressure  is  determined  by 
substituting  rt  for  0  in  equation  (35.),  and  is  represented  by 


*  When  the  pendulum  oscillates  on  knife-edges  a=0,  and  this  expression 
assumes  the  form  of  a  vanishing  fraction,  whose  value  may  be  determined  by 
the  known  rules.  See  the  next  article. 


668  APPENDIX. 


.  .  .  (51). 


g    '  w+(a+h)* 

The  cylinder  will  jump  if  this  expression  be  negative,  that  is,  if 

i 


or,  substituting  and  reducing,  if 


If  the  angular  velocity  «  be  assumed  to  be  that  acquired  in  the  highest 
position  of  the  centre  of  gravity,  0i=rt,  and  cos.  -  81  =  0.  In  this  case, 
therefore  (equation  61.) 


and  there  will  be  a  jump  if  w!>  | .  .  .  (53). 


The  Pendulum  oscillating  on  Knife-edges. 

In  this  case  a  is  evanescent,   and  w=0.     Equations  (31.)  and  (33.) 
become,  therefore, 


s.  0  —  cos.  0,)  __ 

- 


*•  +  # 

Substituting  these  values  of  M  and  K"  in  equation  (30.), 

_       ~Wh*     r  \ 

#  +  #  ]  —  2  (cos'  9  —  cos'  0')  sm'  0  —  cos-  9  sin'  9 1  » 

Y= W  +  F+A7   | (cos-  9  ~  cos'  e^  cos'  °  ~  sin'20  }  ? 

.".  X  =  -^T^-2  (2  cos.  0,  —  3  cos.  0)  sin.  0  ...  (54). 


ROLLING    MOTION   OF   A   CYLINDER.  669 

P\ 
.20— 2008.0008.0,+-!  * ' ' (55)' 


Y  is  a  minimum  when  cos. 9  =  -cos. 0,,  in  which  case 

3 


There  will  therefore  be  a  jump  of  the  pendulum  upon  its  b'earings  at 
each  oscillation  if  the  amplitude  0,  of  the  oscillation  be  such,  that 

1  Z«*  Q 1* 

icos.20,  >  _,  or  cos.20,>riL. 
8  A2  A8 


of  the  falsely -balanced  Carriage-wheel. 

The  theory  of  the  falsely-balanced  carriage- wheel  differs  from  that  of 
the  rolling  cylinder, — 1st,  in  that  the  inertia  of  the  carriage  applied  at  its 
axle  influences  the  acceleration  produced  by  the  weight  of  the  wheel,  as 
its  centre  of  gravity  descends  or  ascends  in  rolling;  and,  2ndly,  in  that 
the  wheel  is  retained  in  contact  with  the  plane  by  the  weight  of  the  car- 
riage. The  first  cause  may  be  neglected,  because  the  displacement  of  the 
centre  of  gravity  is  always  in  the  carriage-wheel  very  small,  and  because 
the  angular  velocity  is,  compared  with  it,  very  great. 

If  W,  represent  that  portion  of  the  weight  of  the  carriage  which  must 
be  overcome  in  order  that  the  wheel  may  jump  (which  weight  is  supposed 
to  be  borne  by  the  plane),  and  if  Y!  be  taken  to  represent  the  pressure 
upon  the  plane,  then  (equation  52.) 

Y1  =  W,+Y=W1  +  w(l ) (57). 

V        ff  ' 

In  order  that  there  may  be  a  jump,  this  expression  must  be  negative, 
or 


670  APPENDIX. 


The  Driving-Wheel  of  a  Locomotive  Engine. 

The  attention  of  engineers  was  some  years  since  directed  to  the  effects 
which  might  result  from  the  false  balancing  of  a  wheel  by  accidents  on 
railways,  which  appeared  to  be  occasioned  by  a  tendency  to  jump  in  the 
driving-wheels  of  the  engines.  The  cranked  axle  in  all  cases  destroys  the 
balance  of  the  driving-wheel  unless  a  counterpoise  be  applied  ;  at  that  time 
there  was  no  counterpoise,  and  the  axle  was  so  cranked  as  to  displace  the 
centre  of  gravity  more  than  it  does  now.  Mr.  GEOKGE  HEATON,  of  Bir- 
mingham, appears  to  have  been  principally  instrumental  in  causing  the 
danger  of  this  false-balancing  of  the  driving-wheels  to  be  understood.  By 
means  of  an  ingenious  apparatus*,  which  enabled  him  to  roll  a  falsely- 
balanced  wheel  round  the  circumference  of  a  table  with  any  given  velocity, 
and  to  make  any  required  displacement  of  the  centre  of  gravity,  he  showed 
the  tendency  to  jump,  produced  even  by  a  very  small  displacement,  to  be 
so  great,  as  to  leave  no  doubt  on  the  minds  of  practical  men  as  to  the 
danger  of  such  displacement  in  the  case  of  locomotive  engines,  and  a  coun- 
terpoise is  now,  I  believe,  always  applied.  To  determine  what  is  the 
degree  of  accuracy  required  in  such  a  counterpoise,  I  have  calculated  from 
the  preceding  formulae  that  displacement  of  the  centre  of  gravity  of  a 
driving-wheel  of  a  locomotive-engine,  which  is  necessary  to  cause  it  to 
jump  at  the  high  velocities  not  unfrequently  attained  at  some  parts  of  the 
journey  of  an  express  train ;  from  such  information  as  I  have  been  able  to 
obtain  as  to  the  dimensions  of  such  wheels,  and  their  weights,  and  those 
of  the  engines  f.  The  weight  of  a  pair  of  driving-wheels,  six  feet  in 
diameter,  with  a  cranked  axle,  varies,  I  am  told,  from  2£  to  3  tons ;  and 
that  of  an  engine  on  the  London  and  Birmingham  Kailway,  when  filled  with 
water,  from  20  to  25  tons.  If  n  represent  the  number  of  miles  per  hour 
at  which  the  engine  is  travelling,  it  may  be  shown  by  a  simple  calculation, 

that  the  angular  velocity,  in  feet,  of  a  six-feet  wheel  is  represented  by  — -=1 
or  by  -n  very  nearly.     In  this  case  we  have,  therefore, — since  "W  represents 

the  weight  of  a  single  wheel  and  its  portion  of  the  axle,  and  "VV,  represents 
the  weight,  exclusive  of  the  driving-wheels,  which  must  be  raised  that 


*  This  apparatus  was  exhibited  by  the  late  Professor  Co WPEK  to  illustrate  his 
Lectures  on  Machinery  at  King's  College.  It  has  also  been  placed  by  General 
MORIN  among  the  apparatus  of  the  Conservatoire  des  Arts  et  Metiers  at  Paris. 

\  I  have  not  included  in  this  calculation  the  inertia  of  the  crank  rods,  of  the 
slide  gearing,  or  of  the  piston  and  piston  rods.  The  effect  of  these  is  to  increase 
the  tendency  to  jiimp  produced  by  the  displacement  of  the  centre  of  gravity 
of  the  wheel ;  and  the  like  effect  is  due  to  the  thrust  upon  the  piston  rod, 
The  discussion  of  these  subjects  does  not  belong  to  my  present  paper. 


ROLLING   MOTION   OF   A   CYLINDER. 


G71 


either  side  of  the  engine  may  jump*,  that  is,  half  the  weight  of  the  engine 
exclusive  of  the  driving-wheels, — W  =  !£  to  H  tons,  W,  =  8f  to  Hi  tons, 

w  =  -TO,  ^  =  32-19084  whence  I  have  made  the  following  calculations  from 
formula  (59.). 


Height  of 
the  engine  in 
tons,  includ- 
ing the  driv- 
ing wheels. 

Weight  of  a 
pair  of  wheels 
with   cranted 
axle,  in  tons. 

Formula  (59.) 
reduced. 

J2,re(,4') 

Displacement  of  the  centre  of  gravity 
of  a  six-feet  driving-wheel  which 
will  cause  a  jump  of  the  wheel  on 
the  rail. 

Eate  of  travelling  in  miles  per  hour. 

«.* 

50. 

60. 

70. 

20 

2-5 
3 

1030-08 

•4128 
•3434 

•2867 
•2384 

•2106 
,  '1751 

n* 

858-4 

«.' 

25 

2-5 

3 

1287.6 

•5150 
•4292 

•3576 
•2908 

•2628 
•2189 

n* 

1073 
ijiiS 

It  appears,  by  formula  (59.),  that  the  displacement  of  the  centre  of 
gravity  necessary  to  produce  a  jump  at  any  given  speed,  is  not  dependent 
on  the  actual  weight  of  the  engine  or  the  wheels,  but  on  the  ratio  of  their 
weights ;  and,  from  the  above  table,  that  when  the  weight  of  the  engine 
and  wheels  is  6$-  times  that  of  the  driving-wheels,  a  displacement  of  2f 
inches  in  the  centre  of  gravity  is  enough  to  create  a  jump  when  the  train 
is  travelling  at  sixty  miles  an  hour,  or  of  two  inches  when  it  is  travelling 
at  seventy  miles ;  this  displacement  varying  inversely  as  the  square  of  the 
velocity  is  less,  other  things  being  the  same,  as  the  square  of  the  diameter 
of  the  wheel  is  less ;  for  the  radius  of  the  wheel  being  represented  by  0, 

the  angular  velocity  is  represented  by  «  =  z-^— •>  and  substituting  this  value, 
formula  (59.)  becomes 


*  It  will  be  observed,  that  the  cranks  being  placed  on  the  axle  at  right  angles 
to  one  another,  when  the  centre  of  gravity  on  the  one  side  is  in  a  favourable  po- 


APPENDIX. 

If  the  weight  "W  of  the  wheel  be  supposed  to  vary  as  the  square  of  its 
diameter  and  be  represented  by  pd*,  this  formula  will  become 


h>(M 


still  showing  the  displacement  of  the  centre  of  gravity  necessary  to  pro- 
duce a  jump  to  diminish  with  the  diameter  of  the  wheel.  These  conclu- 
sions are  opposed  to  the  use  of  light  engines  and  small  driving-wheels; 
and  they  show  the  necessity  of  a  careful  attention  to  the  true  balancing  of 
the  wheels  of  the  carriages  as  well  as  the  driving-wheels  of  the  engine. 
It  does  not  follow  that  every  jump  of  the  wheel  would  be  high  enough  to 
lift  the  edge  of  the  flange  off  the  rail ;  the  determination  of  the  height  of 
the  jump  involves  an  independent  investigation.  Every  jump  nevertheless 
creates  an  oscillation  of  the  springs,  which  oscillation  will  not  of  necessity 
be  completed  when  the  jump  returns  ;  but  as  the  jumps  are  made  alter- 
nately on  opposite  sides  of  the  engine,  it  is  probable  that  they  may,  and 
that  after  a  time  they  will,  so  synchronise  with  the  times  of  the  oscillations, 
as  that  the  amplitude  of  each  oscillation  shall  be  increased  by  every  jump, 
and  a  rocking  motion  be  communicated  to  the  engine  attended  with 
danger. 

Whilst  every  jump  does  not  necessarily  cause  the  wheel  to  run  off  the 
rail,  it  nevertheless  causes  it  to  slip  upon  it,  for  before  the  wheel  jumps 
it  is  clear  that  it  must  have  ceased  to  have  any  hold  upon  the  rail  or  any 
friction. 

The  Slip  of  the  Wheel. 

If /be  taken  to  represent  the  coefficient  of  friction  between  the  surface 
of  the  wheel  and  that  of  the  rail,  the  actual  friction  in  any  position  of  the 
wheel  will  be  represented  by  Y,  /.  But  the  friction  which  it  is  necessary 
the  rail  should  supply,  in  order  that  the  rolling  of  the  wh«el  maybe  main- 
tained, is  X.  It  is  a  condition  therefore  necessary  to  the  wheel  not  slip- 
ping that 


If,  therefore,  taking  the  maximum  value  of     in  any  revolution,  we  find 

that /exceeds  it,  it  is  certain  that  the  wheel  cannot  have  slipped  in  that 
revolution;  whilst  if,  on  the  other  hand, /falls  short  of  it,  it  must  have 

sition  for  jumping,  it  is  in  an  unfavourable  position  on  the  other  side,  so  that 
It  can  only  jump  on  one  side  at  once,  and  the  efforts  on  the  two  sides  alternate- 


EOLLING    MOTION   OF  A   CYLINDER.  673 

slipped.*    The  positions  between  which  the  slipping  will  take  place  con- 
tinually, are  determined  by  solving,  in  respect  to  cos.  0,,  the  equation 


/=?....  (61). 


The  application  of  these  principles  to  the  slip  of  the  carriage-wheel  is 
rendered  less  difficult  by  the  fact,  that  the  value  of  h  is  always  in  that  case 

so  small,  as  compared  with  the  values  of  Tc  and  a,  that  -  may  be  neglected 

a 

in  formulae  (34.)  and  (35.),  as  compared  with  unity.    Those  equations 
then  become 


1 X.2+     7l (62). 

and 

HI+^-[, 


=        --cos.  a+ 


«  (A 

whence  we  obtain 


X 


Assume 


/i  +  !L'W 

( +wj^+cos. 


e     d?u__{  —  j3  (jS  +  C03.  0)  +2  (1  +|8  cos.  &)}  sin. 
~ 


"  d9    (3  +  cos.  0)2     ^2~  03  +  cos.  e  )3 

Now  if  j3>  1,  there  will  be  some  value  of  e  for  which  ^  +  cos.  0  =  0,  and 

therefore  1+3  cos.  0  =  0;  and  since  for  this  value  of  0,  _  =  0,  and  —^ 

ad  ojr 


*  Of  course,  the  slipping,  in  the  case  of  the  driving-wheels  of  a  locomotive, 
is  diminished  by  the  fact,  that  whilst  one  wheel  IB  not  biting  upon  the  rail 
the  other  is. 

43 


APPENDIX. 


= tM     -3it  follows  that  it  corresponds  to  a  maximum  value  of  u,  and 

therefore  of  5. 
Y, 

But  if  (3  <  1,  then  there  is  some  value  of  cos.  0  for  which  |3  +  cos.  0  =  0, 
and  therefore  for  which  u  —  infinity,  which  value  corresponds  therefore  in 

this  case  to  the  maximum  of  5  • 
Y, 

Thus  then  it  appears  that  according  as 


the  maximum  value  of  =•  is  attained  when  cos.  0=  —  j3  or  =  — ^ ;    that 
is,  when 


Y 

In  the  one  case  the  maximum  value  of  y  will  be  infinity,  ....  (67). 

and  in  the  other  case  it  will  be  represented  by  the  formula 


(68). 

< 


In  the  first  case,  i.  e.  when  /3<  1,  the  wheel  will  slip  every  time  that  it 
revolves,  whatever  may  be  the  value  of  f.  In  the  second  case,  or  when 
3  >  1,  it  will  slip  if  /do  not  exceed  the  number  represented  by  formula 
(68.).  The  conditions  (65.)  are  obviously  the  same  with  those  (59.)  which 
determine  whether  there  be  a  jump  or  not,  which  agrees  with  an  obser- 
vation in  the  preceding  article,  to  the  effect,  that  as  the  wheel  must  cease 
to  bite  upon  the  rail  before  it  can  jump,  it  must  always  slip  before  it 
can  jump.  "When  the  conditions  of  slipping  obtain,  one  of  the  wheels 
always  biting  when  the  other  is  slipping,  and  the  slips  of  the  two  wheels 
alternating,  it  is  evident  that  the  engine  will  be  impelled  forwards,  at 
certain  periods  of  each  revolution,  by  one  wheel  only,  and  at  others,  by 
the  other  wheel  only;  and  that  this  is  true  irrespective  of  the  action  of 
the  two  pistons  on  the  crank,  and  would  be  true  if  the  steam  were  thrown 
off.  Such  alternate  propulsions  on  the  two  sides  of  the  train  cannot  but 


DESCENT    UPON    INCLINED    PLANE.  675 

communicate  alternate  oscillations  to  the  buffer-springs,  the  intervals 
between  which  will  not  be  the  same  as  those  between  the  propulsions; 
but  they  may  so  synchronise  with  a  series  of  propulsions  as  that  the 
amplitude  of  each  oscillation  may  be  increased  by  them  until  the  train 
attains  that  fish-tail  motion  with  which  railway  travellers  are  familiar. 
It  is  obvious  that  the  results  shown  here  to  follow  from  a  displacement  of 
the  centres  of  gravity  of  the  driving-wheels,  cannot  fail  also  to  be  pro- 
duced by  the  alternate  action  of  the  connecting  rods  at  the  most  favorable 
driving  points  of  the  crank  and  at  the  dead  points,*  and  that  the  operation 
of  these  two  causes  may  tend  to  neutralize  or  may  exaggerate  one  another. 
It  is  not  the  object  of  this  paper  to  discuss  the  question  under  this  point 
of  view. 


NOTE  F. 

ON  THE  DESCENT  UPON  AN  INCLINED  PLANE  OF  A  BODY  SUBJECT  TO  VARIA- 
TIONS OF  TEMPERATURE,  AND  ON  THE  MOTION  OF  GLACIERS. 

IF  we  conceive  two  bodies  of  the  same  form  and  dimensions  (cubes,  for 
instance),  and  of  the  same  material,  to  be  placed  upon  a  uniform  horizon- 
tal plane  and  connected  by  a  substance  which  alternately  extends  and 
contracts  itself,  as  does  a  metallic  rod  when  subjected  to  variations  of 
temperature,  it  is  evident  that  by  the  extension  of  the  intervening  rod 
each  will  be  made  to  recede  from  the  other  by  the  same  distance,  and, 
by  its  contraction,  to  approach  it  by  the  same  distance.  But  if  they  be 
placed  on  an  inclined  plane  (one  being  lower  than  the  other)  then  when 
by  the  increased  temperature  of  the  rod  its  tendency  to  extend  becomes 
sufficient  to  push  the  lower  of  the  two  bodies  downwards,  it  will  not  have 
become  sufficient  to  push  the  higher  upwards.  The  effect  of  its  exten- 
sion will  therefore  be  to  cause  the  lower  of  the  two  bodies  to  descend 
whilst  the  higher  remains  at  rest.  The  converse  of  this  will  result  from 
contraction ;  for  when  the  contractile  force  becomes  sufficient  to  pull  the 
upper  body  down  the  plane  it  will  not  have  become  sufficient  to  pull 
the  lower  up  it.  Thus,  in  the  contraction  of  the  substance  which  inter- 
venes between  the  two  bodies,  the  lower  will  remain  at  rest  whilst  the 
upper  descends.  As  often,  then,  as  the  expansion  and  contraction  is 
repeated  the  two  bodies  will  descend  the  plane  until,  step  by  step,  they 
reach  the  bottom. 


*  A  slip  of  the  wheel  may  thus  be,  and  probably  is,  produced  at  each  revo- 
lution. 


C7G  APPENDIX. 

Suppose  the  uniform  bar  AB  placed  on  an  inclined  plane,  and  subject 
to  extension  from  increase  of  temperature,  a  por- 
tion XB  will  descend,  and  the  rest  X  A  will  ascend ; 
the  point  X  where  they  separate  being  determined 
by  the  condition  that  the  force  requisite  to  push 
XA  up  the  plane  is  equal  to  that  required  to  push 
XB  down  it. 

Let  AX  =  a?,  AB  =  L,  weight  of  each  linear  unit  =  /*,  t  =  inclination 
of  plane,  $  =  limiting  angle  of  resistance. 

.-.(jix  =  weight  of  AX. 

Now,  the  force  acting  parallel  to  an  inclined  plane  which  is  necessary 
to  push  a  weight  W  up  it,  is  represented  by  W — li^JLyj   and  that  ne- 

COS.  0 

eessary  to  push  it  down  the  plane  by  W—       — .  (Art.  241.) 


COS. 

^. 
sin.  (?  —  Q 


COS.  <f>  COS.  <p 

.-.x  {sin.  (0  +  t)  +  sin.  (0  —  t)}  =  L  sin.  (p  — 
.  •  .  2  x  sin.  0  cos.  t  =  L  sin.  (^  —  t) 


_.      L 

sm.  <j>  cos. 


,  _   f 

=  iL  < 

( 


tan.  t 
tan.  0 


When  contraction  takes  place,  the  converse  of 
the  above  will  be  true.    The  separating  point  X 
will  be  such,  that  the  force  requisite  to  pull  XB  up 
__^________         the  plane  is  equal  to  that  required  to  pull  AX 

down  it.    BX  is  obviously  in  this  case  equal  to  AX 
in  the  other. 

Let  a.  be  the  elongation  per  linear  unit  under  any  variation  of  tempera- 
ture; then  the  distance  which  the  point  B  (fig.  l.)will  be  made  to  descend 
by  this  elongation  =  x.BX 


DESCENT   UPON   INCLINED   PLANE.  677 

If  we  conceive  the  bar  now  to  return  to  its  former  temperature,  con- 
tracting by  the  same  amount  (X)  per  linear  unit;    then  the  point  B 
(fig.  2.)  will  by  this  contraction  be  made  to  ascend  through  the  space 
~ 


Total  descent  I  of  B  by  elongation  and  contraction  is  therefore  determined 
by  the  equation 


To  determine  the  pressure  upon  a  nail  driven 
through  the  rod  at  any  point  P  fastening  it  to  the 
plane. 

It  is  evident,  that  in  the  act  of  extension  the  part  BP  of  the  rod  will 
descend  the  plane  and  the  part  AP  ascend;  and  conversely  in  the  act  of 
contraction ;  and  that  in  the  former  case  the  nail  B  will  sustain  a  pressure 
upwards  equal  to  that  necessary  to  cause  BP  to  descend,  and  a  pressure 
downwards  equal  to  that  necessary  to  cause  PA  to  ascend ;  so  that,  as- 
suming the  pressure  to  be  downwards,  and  adopting  the  same  notation  as 
before,  except  that  AP  is  represented  by  p,  AB  by  a,  and  the  pressure 
upon  the  nail  (assumed  to  be  downwards)  by  P,  we  have  in  the  case  of 
extension 

_      sin.  (0  +  e,)       .        „  sin. 
and  in  the  case  of  contraction, 

p=,(*-p)sin- 

Reducing,  these  formulae  become  respectively, 

p_ J  2p  sin.  0  cos.  i  —  a  sin.  (0  —  i)  >••••••  (8). 

COS.  <f>  (  j 

P  = <  a  sin.  (<p  + 1)  —  2p  sin.  <t>  cos.  »>••.-•  '  (4). 

COS.  0  (  J 


EXAMPLE  OF  THE  DESCENT  OF  THE  LEAD  ON  THE  ROOF  OF  BEISTOL 

CATHEDEAL. 

My  attention  was  first  drawn  to  the  influence  of  variations  in  tempera- 
ture to  cause  the  descent  of  a  lamina  of  metal  resting  on  an  inclined  plane 


678  APPENDIX. 

by  observing,  in  the  autumn  of  1853,  that  a  portion  of  the  lead  which 
covers  the  south  side  of  the  choir  of  Bristol  Cathedral,  which  had  been 
renewed  in  the  year  1851,  but  had  not  been  properly  fastened  to  the  ridgo 
beam,  had  descended  bodily  eighteen  inches  into  the  gutter;  so  that  if 
plates  of  lead  had  not  been  inserted  at  the  top,  a  strip  of  the  roof  of  that 
length  would  have  been  left  exposed  to  the  weather.  The  sheet  of  lead 
which  had  so  descended  measured,  from  the  ridge  to  the  gutter,  19ft.  4in., 
and  along  the  ridge  60ft.  The  descent  had  been  continually  going  on 
from  the  time  the  lead  had  been  laid  down.  An  attempt  made  to  stop  it 
by  driving  nails  through  it  into  the  rafters  had  failed.  The  force  by 
which  the  lead  had  been  made  to  descend,  whatever  it  was,  had  been 
found  sufficient  to  draw  the  nails.*  As  the  pitch  of  the  roof  was  only 
164-°  it  was  sufficiently  evident  that  the  weight  of  the  lead  alone  could  not 
have  caused  it  to  descend.  Sheet  lead,  whose  surface  is  in  the  state  of 
that  used  in  roofing,  will  stand  firmly  upon  a  surface  of  planed  deal  when 
inclined  at  an  angle  of  30°f,  if  no  other  force  than  its  weight  tends  to 
cause  it  to  descend.  The  considerations  which  I  have  stated  in  the  pre- 
ceding articles,  led  me  to  the  conclusion  that  the  daily  variations  in  the 
temperature  of  the  lead,  exposed  as  it  was  to  the  action  of  the  sun  by  its 
southern  aspect,  could  not  but  cause  it  to  descend  considerably,  and  the 
only  question  which  remained  on  my  mind  was,  whether  this  descent 
could  be  so  great  as  was  observed.  To  determine  this  I  took  the  follow- 
ing data : — 
Mean  daily  variation  of  temperature  at  Bristol  in  the 

month  of  August ;   assumed  to  be  the  same  as  at  Leith 

(Kcemtz  Meteorology,  by  Walker,  p.  18.)      -        -        -      8°  21'  Cent. 
Linear  expansion  of  lead  through  100°  Cent.   -        -        -       '0028436. 
Length  of  sheets  of  lead  forming  the  roof  from  the  ridge 

to  the  gutter  -        - 232  inches. 

Inclination  of  roof 16°  32'. 

Limiting  angle  of  resistance  between  sheet  lead  and  deal  -      80° 

Whence  the  mean  daily  descent  of  the  lead,  in  inches,  in  the  month  of 
August,  is  determined  by  equation  (2.)  to  be 


*  The  evil  was  remedied  by  placing  a  beam  across  the  rafters,  near  the  ridge, 
and  doubling  the  sheets  round  it,  and  fixing  their  ends  with  spike-nails. 

•j-  This  may  easily  be  verified.  I  give  it  as  the  result  of  a  rough  experiment 
of  my  own.  I  am  not  acquainted  with  any  experiments  on  the  friction  of  lead 
made  with  sufficient  care  to  be  received  as  authority  in  this  matter.  The 
friction  of  copper  on  oak  has,  however,  been  determined  by  General  MORIN 
(see  a  table  in  the  preceding  part  of  this  work)  to  be  0'62,  and  its  limiting  angle 
of  resistance  31°  48' ;  so  that  if  the  roof  of  Bristol  Cathedral  had  been  inclined 
at  31°  instead  of  16",  and  had  been  covered  with  sheets  of  copper  resting  on 
oak  boards,  instead  of  sheets  of  lead  resting  on  deal,  the  sheeting  would  not 
have  slipped  by  its  weight  only. 


DESCENT    UPON   INCLINED   PLANE.  679 


Z=-027848  inches. 

This  average  daily  descent  gives  for  the  whole  month  of  August  a  descent 
of  -863288.  If  the  average  daily  variation  of  temperature  of  the  month 
of  August  had  continued  throughout  the  year,  the  lead  would  have 
descended  10-19148  inches  every  year.  And  in  the  two  years  from 
1851  to  1853  it  would  have  descended  20-38296  inches.  But  the  daily 
variations  of  atmospheric  temperature  are  less  in  the  other  months  of  the 
year  than  in  the  month  of  August.  For  this  reason,  therefore,  the  cal- 
culation is  in  excess.  For  the  following  reasons  it  is  in  defect:  —  1st., 
The  daily  variations  in  the  temperature  of  the  lead  cannot  but  have  been 
greater  than  those  of  the  surrounding  atmosphere.  It  must  have  been 
heated  above  the  surrounding  atmosphere  by  radiation  from  the  sun  in 
the  day-time,  or  cooled  below  it  by  radiation  into  space  at  night.  2ndly., 
One  variation  of  temperature  only  has  been  assumed  to  take  place  every 
twenty-four  hours,  viz.  that  from  the  extreme  heat  of  the  day  to  the 
extreme  cold  of  the  night;  whereas  such  variations  are  notoriously  of 
constant  occurrence  during  the  twenty-four  hours.  Each  cannot  but  have 
caused  a  corresponding  descent  of  the  lead,  and  their  aggregate  result 
cannot  but  have  been  greater  than  though  the  temperature  had  passed 
uniformly  (without  oscillations  backwards  and  forwards)  from  one  extreme 
to  the  other. 

These  considerations  show,  I  think,  that  the  causes  I  have  assigned  are 
sufficient  to  account  for  the  fact  observed.  They  suggest,  moreover,  the 
possibility  that  results  of  importance  in  meteorology  may  be  obtained 
from  observing  with  accuracy  the  descent  of  a  metallic  rod  thus  placed 
upon  an  inclined  plane.  That  descent  would  be  a  measure  of  the  aggre- 
gate of  the  changes  of  temperature  to  which  the  metal  was  subjected 
during  the  time  of  observation.  As  every  such  change  of  temperature  is 
associated  with  a  corresponding  development  of  mechanical  action  under 
the  form  of  work,*  it  would  be  a  measure  of  the  aggregate  of  such  changes 
and  of  the  work  so  developed  during  that  period.  And  relations  might  be 
found  between  measurements  so  taken  in  different  equal  periods  of  time 
—  successive  years  for  instance  —  tending  to  the  development  of  new 
meteorological  laws. 


*  Mr.  JOULE  has  shown  (Phil.  Trans.,  1850,  Part  I.)  that  the  quantity  of  heat 
capable  of  raising  a  pound  of  water  by  1°  Fah.  requires  for  its  evolution  772 
units  of  work. 


680  APPENDIX. 

THE  DESCENT  OF  GLAOIEES. 

The  following  are  the  results  of  recent  experiments  *  on  the  expansion 
of  ice : — 

Linear  Expansion  of  Ice  for  am,  Interval  of  100°  of  the  Centigrade 
Thermometer. 

0-00524    Schumacher. 

0-00513     Pohrt. 

0-00518    Moritz. 

Ice,  therefore,  has  nearly  twice  the  expansibility  of  lead ;  so  that  a 
sheet  of  ice  would,  under  similar  circumstances,  have  descended  a  plane 
similarly  inclined,  twice  the  distance  that  the  sheet  of  lead  referred  to  in 
the  preceding  article  descended.  Glaciers  are,  on  an  increased  scale, 
sheets  of  ice  placed  upon  the  slopes  of  mountains,  and  subjected  to 
atmospheric  variations  of  temperature  throughout  their  masses  by  varia- 
tions in  the  quantity  and  the  temperature  of  the  water,  which,  flowing 
from  the  surface,  everywhere  percolates  them.  That  they  must  from  this 
cause  descend  into  the  valleys,  is  therefore  certain.  That  portion  of  the 
Mer  de  Glace  of  Chamouni  which  extends  from  Montanvert  to  very  near 
the  origin  of  the  Glacier  de  Lechaud  has  been  accurately  observed  by 
Professor  James  Forbes.t  Its  length  is  22,600  feet,  and  its  inclination 
varies  from  4°  19'  22"  to  5°  5'  53".  The  Glacier  du  Geant,  from  the 
Tacul  to  the  Col  du  Geant,  Professor  Forbes  estimates  (but  not  from  his 
own  observations,  or  with  the  same  certainty)  to  be  24,700  feet  in  length, 
and  to  have  a  mean  inclination  of  8°  46'  40". 

According  to  the  observations  of  De  Saussure,  the  mean  daily  range 
of  Reaumur's  thermometer  in  the  month  of  July,  at  the  Col  du  Geant,  is 
4° -257},  and  at  Chamouni  10° -09 2.  The  resistance  opposed  by  the 
rugged  channel  of  a  glacier  to  its  descent  cannot  but  be  different  at  dif- 
ferent points,  and  in  respect  to  different  glaciers.  The  following  passage 
from  Professor  Forbes's  work  contains  the  most  authentic  information  I 
am  able  to  find  on  this  subject.  Speaking  of  the  Glacier  of  la  Brenva 
he  says : — u  The  ice  removed,  a  layer  of  fine  mud  covered  the  rock,  not 
composed,  however,  alone  of  the  clayey  limestone  mud,  but  of  sharp  sand 
derived  from  the  granitic  moraines  of  the  glacier,  and  brought  down  with 
it  from  the  opposite  side  of  the  valley.  Upon  examining  the  face  of  the 
ice  removed  from  contact  with  the  rock,  we  found  it  set  all  over  with 
sharp  angular  fragments,  from  the  size  of  grains  of  sand  to  that  of  a 
cherry,  or  larger,  of  the  same  species  of  rook,  and  which  were  so  firmly 


Vide  Archir.  £,  Wissenschaftl.  Kunde  v.  Russland,  Bd.  vii  s.  883. 
Travels  through  the  Alps  of  Savoy.     Edinburgh,  1853. 
Quoted  by  Professor  FORBES,  p.  231. 


DESCENT    OF   GLACIERS.  681 

fixed  in  the  ice  as  to  demonstrate  the  impossibility  of  such  a  surface  being 
forcibly  urged  forwards  without  sawing  any  comparatively  soft  body 
which  might  be  below  it.  Accordingly,  it  was  not  difficult  to  discover  in 
the  limestone  the  very  grooves  and  scratches  which  were  in  the  act  of 
being  made  at  the  time  by  the  pressure  of  the  ice  and  its  contained  frag- 
ments of  stone."  (Alps  of  the  Savoy,  pp.  203 — 4.)  It  is  not  difficult 
from  this  description  to  account  for  the  fact  that  small  glaciers  are  some- 
times seen  to  lie  on  a  slope  of  30°  (p.  35.).  The  most  probable  supposition 
would  indeed  fix  the  limiting  angle  of  resistance  between  the  rock  and 
the  under  surface  of  the  ice  set  all  over,  as  it  is  described  to.  be,  with 
particles  of  sand  and  small  fragments  of  stone,  at  about  30° ;  that  being 
nearly  the  slope  at  which  smooth  surfaces  of  calcareous  stone  will  rest  on 
one  another.  If  we  take  then  30°  to  be  the  limiting  angle  of  resistance 
between  the  under  surface  of  the  Mer  de  Glace  and  the  rock  on  which  it 
rests,  and  if  we  assume  the  same  mean  daily  variation  of  temperature 
(4-257  Keaumur,  or  5'321  Centigrade)  to  obtain  throughout  the  length 
of  the  Glacier  du  Geant,  which  De  Saussure  observed  in  July,  at  the 
Col  du  Geant ;  if,  further,  we  take  the  linear  expansion  of  ice  at  100° 
Centigrade  to  be  that  ('00524)  which  was  determined  by  the  experiments 
of  Schumacher,  and,  lastly,  if  we  assume  the  Glacier  de  Geant  to  descend 
as  it  would  if  its  descent  were  unopposed  by  its  confluence  with  the 
Glacier  de  Lechaut;  we  shall  obtain,  by  substitution  in  equation  (2.) 
for  the  mean  daily  descent  of  the  Glacier  du  Geant  at  the  Tacul,  the 
formula 

1  =  24700  X  5-321  x '^24      tan- 8°  46' 


100         tan.  30° 
1=  1-8395  feet. 

The  actual  descent  of  the  glacier  in  the  centre  was  T5  feet.  If  the 
Glacier  de  Lechaut  descended,  at  a  mean  slope  of  5°,  singly  in  a  sheet  of 
uniform  breadth  to  Montanvert  without  receiving  the  tributary  glacier  of 
the  Talefre,  or  uniting  with  the  Glacier  du  Geant,  its  diurnal  descent  would 
be  given  by  the  same  formula,  and  would  be  found  to  be  -95487  feet. 
Eeasoning  similarly  with  reference  to  the  Glacier  du  Geant ;  supposing  it 
to  have  continued  its  course  singly  from  the  Col  du  Geant  to  Montanvert 
without  confluence  with  the  Glacier  de  Lechaut,  its  length  being  40,420 
feet,  and  its  mean  inclination  6°  53',  its  mean  diurnal  motion  I  at  Montan- 
vert would,  by  formula  (2.)  have  been  2'3564*  feet.  The  actual  mean 
daily  motion  of  the  united  glaciers,  between  the  1st  and  the  28th  July,  was 
at  Montanvert,f 


*  On  the  1st  of  July  the  centre  of  the  actual  motion  of  the  Mer  de  Glace  at 
Montanvert  was  2*25  feet. 

f  Forbes'  "  Alps  of  Savoy,"  p.  140. 


682  APPENDIX. 

Near  the  side  of  the  glacier    -    -     1-441  feet. 
Between  the  side  and  the  centre  -     1  '750     " 
Near  the  centre 2-141     " 

The  motion  of  the  Glacier  de  Lechaut  was  therefore  accelerated  by  their 
confluence,  and  that  of  the  Glacier  du  Geant  retarded.  The  former  is 
dragged  down  by  the  latter. 

I  have  had  the  less  hesitation  in  offering  this  solution  of  the  mechanical 
problem  of  the  motion  of  glaciers,  as  those  hitherto  proposed  are  con- 
fessedly imperfect.  That  of  De  Saussure,  which  attributes  the  descent  of 
the  glacier  simply  to  its  weight,  is  contradicted  by  the  fact  that  isolated 
fragments  of  the  glacier  stand  firmly  on  the  slope  on  which  the  whole 
nevertheless  descends.  It  being  obvious  that  if  the  parts  would  remain  at 
rest  separately  on  the  bed  of  the  glacier,  they  would  also  remain  at  rest 
when  united. 

That  of  Professor  J.  Forbes,  which  supposes  a  viscous  or  semirfiuid 
structure  of  the  glacier,  is  not  consistent  with  the  fact  that  no  viscosity  is 
to  be  traced  in  its  parts  when  separated.  They  appear  as  solid  fragments, 
and  they  cannot  acquire  in  their  union  properties  in  this  respect  which 
individually  they  have  not. 

Lastly,  the  theory  of  Oharpentier,  which  attributes  the  descent  of  the 
glacier  to  the  daily  congelation  of  the  water  which  percolates  it,  and  the 
expansion  of  its  mass  consequent  thereon,  whilst  it  assigns  a  cause  which, 
so  far  as  it  operates,  cannot,  as  I  have  shown,  but  cause  the  glacier  to 
descend,  appears  to  assign  one  inadequate  to  the  result ;  for  the  congelation 
of  the  water  which  percolates  the  glacier  does  not,  according  to  the  obser- 
vations of  Professor  Forbes,*  take  place  at  all  in  summer  more  than  a  few 
inches  from  the  surface.  Nevertheless,  it  is  in  the  summer  that  the  daily 
motion  of  the  glacier  is  the  greatest. 

The  following  remarkable  experiment  of  Mr.  Hopkins  of  Cambridge,! 
which  is  considered  by  him  to  be  confirmatory  of  the  sliding  theory  of 
De  Saussure  as  opposed  to  De  Oharpentier's  dilatation  theory,  receives 
a  ready  explanation  on  the  principles  which  I  have  laid  down  in  this 
note.  It  is  indeed  a  necessary  result  of  them.  Mr.  Hopkins  placed  a 
mass  of  rough  ice,  confined  by  a  square  frame  or  bottomless  box,  upon 
a  roughly  chiselled  flag-stone,  which  he  then  inclined  at  a  small  angle ; 
and  found  that  a  slow  but  uniform  motion  was  produced,  when  even  it 
was  placed  at  an  inconsiderable  slope.  This  motion,  which  Mr.  Hopkins 
attributed  to  the  dissolution  of  the  ice  in  contact  with  the  stone,  would, 
I  apprehend,  have  taken  place  if  the  mass  had  been  of  lead  instead  of  ice ; 


*  "Travels  in  the  Alps,"  J>.  413. 

f  I  have  quoted  the  following  account  of  it  from  Professor  Forbes's  book, 
p.  419. 


DIMENSIONS   OF   A   BUTTRESS.  (>S3 

and  it  would  have  been  but  about  half  as  fast,  because  the  linear  expan- 
sion of  lead  is  only  about  half  that  of  ice. 


NOTE  G. 
THE  BEST  DIMENSIONS  OF  A  BTJTTBESS. 

IF  mi  (Art.  299.)  represent  the  modulus  of  stability  of  the  portion  AG  of 
the  wall,  it  may  be  shown,  as  before,  that 

P{  (&,  —  A2)  sin.  a  —  (I —  a3  —  m,)cos.  a}  =  (£a,  —  w,)  (A,  —  A2)a,^t ; 

.  • .  P{(^i  —  A2)sin.  a  —  (I  —  a2)cos.  a} 
==  K&i  —  ^2)^1  V — wi{P  C08- »  +  (^i  —  A*X/*} 

If  ml=m^  the  stability  of  the  portion  AG  of  the  structure  is  the  same 
with  that  of  the  whole  AC ;  an  arrangement  by  which  the  greatest 
strength  is  obtained  with  a  given  quantity  of  material  (see  Art.  388.). 
This  supposition  being  made,  and  m  eliminated  between  the  above  equa- 
tion and  equation  (392.),  that  relation  between  the  dimensions  of  the 
buttress  and  those  of  the  wall  which  is  consistent  with  the  greatest 
economy  of  the  material  used  will  be  determined.  The  following  is  that 
relation : — 

,  +  ~afhz}  —  P  (A,  sin.  a  —  I  cos.  a) 


P  cos.  a  +  nt(a A  +  ~«A) 


— AsK* — P{(Ai — A»)  sin,  a  —  (I— a,) cos,  a} 
P  cos.  a  +  ^  (hi  —  h3) 

It  is  necessary  to  the  greatest  economy  of  the  material  of  the  Gothic 
buttress  (Art.  301.)  that  the  stability  of  the  portions  Qa  and  Q5,  upon 
their  respective  bases  ao  and  fo,  should  be  same  with  that  of  the  whole 
buttress  on  its  base  EC.  If,  in  the  preceding  equation,  hl— h3  be 
substituted  for  A,,  and  7i2  —  h3  for  A2,  the  resulting  equation,  together 
with  that  deduced  as  explained  in  the  conclusion  of  Art.  301.,  will  deter- 
mine this  condition,  and  will  establish  those  relations  between  the  dimen- 
sions of  the  several  portions  of  the  buttress  which  are  consistent  with  the 
greatest  economy  of  the  material,  or  which  yield  the  greatest  strength  to 
the  structure  from  the  use  of  a  given  quantity  of  material. 


684: 


APPENDIX. 


NOTE  H. 
DIMENSIONS  OP  THE  TEETH  OF  WHEELS. 

THE  following  rules  are  extracted  from  the  work  of  M.  Morin,  entitled 
Aide  Memoire  de  Mecanique  Pratique  : — If  we  represent  by  a  the  width 
in  parts  of  a  foot  of  the  tooth  measured  parallel  to  the  axis  of  the  wheel, 
and  by  I  its  breadth  or  thickness  measure*  parallel  to  the  plane  oi 
rotation  upon  the  pitch  circle ;  then,  the  teeth  being  constantly  greased, 
the  relation  of  a  and  5  should  be  expressed,  when  the  velocity  of  the  pitch 
circle  does  not  exceed  5  feet  per  second,  by  a  =  45 ;  when  it  exceeds 
5  feet  per  second,  by  a  =  55 :  if  the  wheels  are  constantly  exposed  to  wet, 
by  a=6&. 

These  relations  being  established,  the  width  or  thickness  of  the  tooth 
will  be  determined  by  the  formulae  contained  in  the  columns  of  the  follow- 
ing table: — 


Material 

French  measures,  cents 
and  kils. 

English  measures,  feet 
and  pounds. 

Cast  iron  ... 
Brass  ... 
Hard  wood 

b=-I05V~P' 
6—13H/F 
6  =  U5|/P 

b  =  '00231  9  V~? 
6=  -002894  f/F 
6=  -003203  4/F 

Assuming  that  when  the  teeth  are  carefully  executed  the  space  between 
the  teeth  should  be  T^th  greater  than  their  thickness,  and  y'^th  greater 
when  the  least  labor  is  bestowed  on  them,  the  values  of  the  pitch  T  will 
in  these  two  cases  be  represented  by  6(2  +  Ts)  and  6(2 +  TV),  or  by  2-0676 
and  2 '15.  Substituting  in  these  expressions  the  values  of  6  given  by  the 
formulas  of  the  preceding  table,  then  determining  from  the  resulting 
values  of  c  (see  equation  233.)  the  corresponding  values  of  the  coefficient 
0  (see  equation  234.),  the  following  table  is  obtained : — 


Material. 

Value  of  o  (equation  238.). 

Value  of  C  (equation  234.). 

For  teeth  of 
the  best  work- 
manship. 

For  teeth  of 
interior  work- 
manship. 

For  teeth  of 
the  hest  work- 
manship. 

For  teeth  of 
inferior  work- 
manship. 

Cast  iron  - 
Brass 
Hard  wood 

•004795 
•005982 
•006621 

•004870 
•006077 
•006726 

0-912 
1-057 
1-131 

0-922 
1  -068 
1-143 

TRACTION   OF   CARRIAGES.  685 

The  following  are  the  pitches  commonly  in  use  among  mechanics : — 
in.      in.       in.       in.       in.     in.      in. 
1,     li,     li,     H,     2,     2i,     3. 

Prof.  Willis  considers  the  following  to  be  sufficient  below  inch  pitch : — 

in.     in.     in.     in.     in. 

i,    t,    *,    *,    *- 

Having,  therefore,  determined  the  proper  pitch  to  be  given  to  the  tooth 
from  formula  234.,  the  nearest  pitch  is  to  be  taken  from  the  above  series 
to  that  thus  determined. 


NOTE  I. 
EXPERIMENTS  OF  M.  MOKIN  ON  THE  TRACTION  OF  OAEBIAGES. 

THE  following  are  among  the  general  results  deduced  by  M.  Morin  from 
his  experiments : — 

1.  The  traction  is  directly  proportional  to  the  load,  and  inversely  pro- 
portional to  the  diameter  of  the  wheel. 

2.  Upon  a  paved  or  a  hard  Macadamised  road,  the  resistance  is  independ- 
ent of  the  width  of  the  tire  when  it  exceeds  from  3  to  4  inches. 

3.  At  a  walking  pace  the  traction  is  the  same,  under  the  same  circum- 
stances, for  carriages  with  springs  and  without  them. 

4.  Upon  hard  Macadamised  and  upon  paved  roads  the  traction  increases 
with  the  velocity ;  the  increments  of  traction  being  directly  proportional 
to  the  increments  of  velocity  above  the  velocity  3'28  feet  per  second,  or 
about  2i  miles  per  hour.     The  equal  increment  of  traction  thus  due  to 
each  equal  increment  of  velocity  is  less  as  the  road  is  more  smooth,  and 
the  carriage  less  rigid  or  better  hung. 

4.  Upon  soft  roads  of  earth,  or  sand  or  turf,  or  roads  fresh  and  thickly 
graveled,  the  traction  is  independent  of  the  velocity. 

5.  Upon  a  well-made  and  compact  pavement  of  hewn  stones  the  traction 
at  a  walking  pace  is  not  more  than  three-fourths  of  that  upon  the  beat 
Macadamised  road  under  similar  circumstances;  at  a  trotting  pace  it 
is  equal  to  it. 

6.  The  destruction  of  the  road  is  in  all  cases  greater  as  the  diameters 
of  the  wheels  are  less,  and  it  is  greater  in  carriages  without  than  with 
springs. 


APPENDIX. 

NOTE  K. 
ON  THE  STKENGTH  OF  COLUMNS. 

MB.  HODGKINSON  has  obligingly  communicated  the  following  observations 
on  Art.  430.  :— 

1.  The  reader  must  be  made  to  understand  that  the  rounding  of  the 
ends  of  the  pillars  is  to  make  them  moveable  there,  as  if  they  turned  by 
means  of  a  universal  joint ;  and  the  flat-ended  pillars  are  conceived  to  be 
supported  in  every  part  of  the  ends  by  means  of  flat  surfaces,  or  otherwise 
rendering  the  ends  perfectly  immoveable. 

2.  The  coefficient  (13)  for  hollow  columns  with  rounded  ends  is  deduced 
from  the  whole  of  the  experiments  first  made,  including  some  which  were 
very  defective  on  account  of  the  difficulty  experienced  in  the  earlier 
attempts  to  cast  good  hollow  columns  so  small  as  were  wanted.     The 
first  castings  were  made  lying  on  their  side ;  and  this,  notwithstanding 
every  effort,  prevented  the  core  being  in  the  middle ;  some  of  the  columns 
were  reduced,  too,  in  thickness,  half  way  between  the  middle  and  the 
ends,  and  near  to  the  ends,  and  this  slightly  reduced  the  strength.     These 
causes  of  weakness  existed  much  more  among  the  pillars  with  rounded 
ends  than  those  with  flat  ones ;  they  are  alluded  to  in  the  paper  (Art.  47.). 
Had  it  not  been  for  them,  the  coefficient  (13)  would,  I  conceive,  have 
been  equal  to  that  for  solid  pillars  (or  14- 9). 

3.  The  fact  of  long  pillars  with  flat  ends  being  about  three  times  as 
strong  as  those  of  the  same  dimensions  with  rounded  ends  is,  I  conceive, 
well  made  out,  in  cast  iron,  wrought  iron,  and  timber;  you  have,  how- 
ever, omitted  it,  being  perhaps  led  to  do  it  through  the  low  value  of  the 
coefficient  (13)  above  mentioned. 

The  same  may  be  mentioned  with  respect  to  the  near  approach  in 
strength  of  long  pillars  with  flat  ends,  and  those  of  half  the  length  with 
rounded  ends.  It  may  be  said  that  the  law  of  the  1*7  power  of  the  length 
would  nearly  indicate  the  latter  ;  but  this  last,  and  the  other  powers  3 '76 
and  3'55,  are  only  approximations,  and  not  exactly  constant,  though 
nearly  so,  and  I  do  not  know  whether  the  other  equal  quantities  are  not, 
with  some  slight  modifications,  physical  facts. 

4.  The  strength  of  pillars  of  similar  form  and  of  the  same  materials 
varies  as  the  1*865  power,  or  near  as  the  square  of  their  like  linear 
dimensions,  or  as  the  area  of  their  cross  section. 


COMPLETE   ELLIPTIC   FUNCTIONS. 


687 


TABLE  I. 

The  Numerical  Values  of  COMPLETE  Elliptic  Functions  of  the  FIRST  and 
SECOND  Orders  for  Values  of  the  Modulus  k  corresponding  to  each  Degree 
of  the  Angle  sin.— '&. 


Sin.—1*. 

Fi. 

Ei. 

Sin.—  ik. 

F,. 

E,. 

0° 
1 
2 
3 

4 
5 

1-57079 
1-57091 
1-57127 
1-57187 
1-57271 
1-57379 

1-57079 
1-57067 
1-57031 
1-56972 
1-56888 
1-56780 

46 

47 
48 
49 
50 

1-86914 

1-88480 
1-90108 
1-91799 
1-93558 

1-34180 
1-33286 
1  -32384 
1-31472 
1-30553 

51 
52 
53 
54 
55 

1-95386 
1-97288 
1-99266 
2-01326 
2-03471 

1-29627 
1-28695 
1-27757 
1-26814 
1-25867 

6 
7 
8 
9 
10 

1-57511 

1-57667 
1-57848 
1-58054 
1*58284 

1-56649 
1-56494 
1-56316 
1-56114 
1-55888 

56 
67 
58 
59 
60 

2-05706 
2-08035 
2-10465 
2-13002 
2-15651 

1-24918 
1-23966 
1-23012 
1-22058 
1-21105 

11 
12 
13 
14 
15 

1-58539 
1-58819 
1-59125 
1-59456 

1-59814 

1-55639 
1-55368 
1-550>J8 
1-54755 
1-54415 

61 
62 
63 
64 
65 

2-18421 
2-21319 
2-24354 
2-27537 
2-30878 

1-20153 
1-19204 
1-18258 
1-17317 
1-16382 

16 
17 
18 
19 
20 

1-60197 
1-60608 
1-61045 
1-61510 
1-62002 

1-54052 
1-53666 
1-53259 
1  -52830 
1-52379 

66 
67 
68 
69 
70 

2-34390 
2-38087 
2-41984 
2-46099 
2-50455 

1-15454 
1-14534 
1-13624 
1-12724 
1-11837 

21 
22 
23 
24 
25 

1-62523 
1-63072 
1-63651 
1  -64260 
1-64899 

1-51907 
1-51414 
1-50900 
1-50366 
1-49811 

71 

72 
73 
74 
75 

2-55073 
2-59981 
2-65213 
2-70806 
2-76806 

1-10964 
1-10106 
1  -09265 
1  -08442 
1-07640 

26 

27 

28 
29 
30 

1  -65569 
1-66271 
1-67005 
1-67773 
1-68575 

1-49236 
1-48642 
1-48029 
1-47396 
1-46746 

76 

77 
78 
79 
80 

2*83267 
2-90256 
2-97856 
3-06172 
3-15338 

1-06860 
1-06105 
1-05377 
1-04678 
1-04011 

31 
32 
33 
34 
35 

1-69411 
1-70283 
1-71192 
1-72139 
1-73124 

1-46077 
1-45390 
1-44686 
1-43966 
1-43229 

81 

82 
83 

84 
85 

3-25530 
3-36986 
3-50042 
3-65186 
3-83174 

1-03378 
1-02784 
1-02281 
1-01723 
1-01266 

3''> 
37 

3S 
39 
40 

1-74149 
1-75216 
1-76325 
1-77478 
I  -7807  8 

1-42476 
1-41707 
1-40923 
1-40125 
1-39314 

88 

87 
88 
89 

4-05275 
4-33865 
4-74271 
5-43490 

1-00864 
1-00525 
1-00258 
1-00075 

41 
42 
43 
44 

45 

1-79922 
l-R121o 
1-82500 
1-83956 
1-85407 

1-38488 
1-37650 
1-36799 
1-35937 
1  -35064 

4PPENDIX. 


THE  TABLES  OF  M.  GAEIDEL. 

TABLE  II. 

Showing  the  Angle  of  Rupture  ¥  of  an  Arch  whose  Loading  is  of  the  same 
Material  with  its  Voussoirs,  and  whose  Extrados  is  inclined  at  a  given 
Angle  to  the  Horizon.  (See  Art.  344.)' 

a  =  ratio  of  lengths  of  voussoirs  to  radius  of  intrados. 

c  =  ratio  of  depth  of  load  over  crown  to  radius  of  intrados,  so  thut 

e  =  /3(l+a).     (Art.  838.) 
t  =  inclination  of  extrados  to  horizon. 


a 

c=0 

o=0-l 

c=0-2 

c=0-8 

c=0-4 

c=0-5 

c=l-0 

0-05 

68-0° 

69-19° 

54-04° 

51-15° 

49-35° 

48-20° 

45-74° 

o-io 

65-4 

60-48 

57-70 

56-01 

54-93 

54-17 

52-34 

0-15 

64-0 

61-3 

59-7 

58-69 

58-0 

57-49 

56-21 

0-20 

631 

61-7 

60-88 

60-30 

59-90 

59-60 

58-80 

0-25 

62-24 

61-76 

61-44 

61-22 

61-05 

60-94 

60-59 

0-30 

61-3 

61-42 

61-54 

61-60 

61-66 

61-67 

61-81 

0-35 

60-17 

60-80 

61-21 

61-54 

61-78 

61-98 

62-56 

0-40 

58-8 

69-8 

60-52 

61-05 

61-48 

61-67 

62-9 

0-45 

67-32 

58-53 

59-45 

60-19 

60-80 

61-28 

62-85 

0-50 

65-63 

56-97 

58-09 

58-98 

59-72 

60-34 

62-40 

=  T°  30'. 


a 

c=0 

c=0-l 

c=0-2 

c=0-3 

c=0-4 

c=0-5| 

c=l-0 

005 

68-3° 

57-3° 

51-69° 

48-61° 

47-84° 

46-11° 

44-85° 

0-10 

64-3 

58-68 

55-95 

54-52 

53-64 

53-03 

51-68 

0-15 

62-43 

59-67 

58-33 

57-55 

57-00 

56-61 

55-66 

0-20 

61-48 

60-42 

59-72 

59-35 

59-07 

58-87 

58-29 

0-25 

60-75 

60-55 

60-44 

60-38 

60-33 

60-21 

60-17 

0-30 

60-09 

60-49 

60-77 

60-95 

61-08 

61-18 

61-48 

0-35 

59-27 

60-12 

60-62 

61-02 

61-33 

61-59 

62-31 

0-40 

58-25 

59-33 

60-11 

60-72 

61-18 

61-57 

62-7 

0-45 

67-11 

58-35 

59-29 

60-05 

60-67 

61-16 

62-78 

0-50          55-82 

57-13 

58-21 

59-08 

59-81 

60-41 

62-45 

ANGLE   OF   RUPTURE   OF    AN   AKCH, 


689 


=  15'. 


a 

0=0 

o=0-l 

c=0-3 

c=0-8 

c=0-4 

c=0-5 

o-l-O 

0-05 

64-8° 

50-5° 

46-95° 

45-69° 

45-03° 

44-67° 

43-9° 

O'lO 

59-3 

55-07 

53-34 

52-47 

51-99 

51-69 

50-93 

0-15 

59-08 

57-32 

56-65 

56-05 

55-75 

55-55 

55-05 

0-20 

59-06 

68-60 

58-35 

58-20 

58-10 

68-02 

57-84 

0-25 

59-05 

59-28 

59-42 

o9-53 

59-60 

59-65 

59-79 

0-30 

58-90 

59-57 

59-98 

60-26 

60-48 

60-66 

61-15 

0-35 

58-53 

59-41 

60-09 

60-57 

60-93 

61-17 

62-0 

0-40 

57-99 

59-08 

59-87 

60-48 

60-95 

61-36 

62-6 

0-45 

67-26 

58-43 

59-34 

60-06 

60-67 

61-15 

62-7 

0-50 

56-38 

57-61 

58-58 

59-36 

60-06 

60-64 

62-5 

=  22°  30'. 


a 

c=0 

c=0-l 

0=0-2 

c=0-8 

c=0-4 

o=0-5 

0=1-0 

0-05 

36-1° 

41-2° 

42-0° 

42-3° 

42-6° 

42-7° 

42-9° 

o-io 

50-5 

50-3 

50-19 

5017 

50-14 

50-13 

50-11 

0-15 

54-25 

54-31 

54-35 

54-35 

54-36 

54-36 

54-38 

0-20 

56-17 

56-60 

56-82 

56-95 

57-04 

57-11 

57-28 

0-25 

57-27 

57-93 

58-33 

58-61 

58-79 

58-95 

59-33 

0-30 

57-85 

58-68 

59-23 

59-60 

59-93 

60-16 

60-83 

Q-35 

68-07 

59-01 

59-70 

60-21 

60-61 

60-91 

61-85 

0-40 

58-02 

59-02 

69-79 

60-38 

60-87 

61-25 

62-2 

0-45 

57-74 

58-78 

69-60 

60-26 

60-82 

61-27 

62-7 

0-50 

57-30 

58-31 

59-16 

5988 

60-47 

61-00 

62-9 

=  30°. 


• 

c=0 

o=0-l 

0=0-3 

c=0-8 

c=0-4 

0=0-5 

0=1-0 

0-05 

31-3° 

36-2° 

38-4° 

39-57° 

40-28° 

40-77° 

41-9° 

o-io 

43-3 

46-06 

47-25 

47-90 

48-30 

48-59 

49-24 

015 

50-07 

51-46 

52-18 

52-63 

62-94 

53-14 

53-68 

0-20 

53-66 

54-69 

55-27 

55-67 

55-96 

56-16 

56-72 

0-25 

55-80 

56-72 

57-30 

57-72 

58-01 

58-23 

58-89 

0-30 

57-13 

58-01 

58-62 

59-06 

59-40 

59-69 

60-48 

0-35 

57-93 

58-80 

59-43 

59-94 

60-33 

60-66 

61-64 

0-40 

58-33 

59-20 

59-89 

60-42 

60-87 

61-23 

62-39 

0-45 

58-47 

59-33 

60-03 

6061 

61-08 

61-48 

62-87 

0-50 

58-38 

59-22 

59-93 

60-53 

61-03 

61-47 

63-0 

44 


690 


APPENDIX. 


=  37°  30'. 


a 

0=0 

0=01 

e=0-2 

o=0-3 

c=0-4 

0=0-5 

0=1-0 

0-05 

3M° 

34-3° 

36-28° 

37-59° 

38-48° 

39-16° 

40-82° 

o-io 

40-98 

43-59 

45-09 

46-01 

46-67 

47-14 

48-35 

0-15 

47-71 

49-40 

50-43 

51-12 

51-61 

51-96 

52-93 

0-20 

52-01 

52-23 

54-01 

54-54 

54-94 

55-24 

56-10 

0-25 

54-87 

55-80 

56-45 

56-94 

57-29 

57-59 

58-41 

0-30 

56-77 

57-o8 

58-16 

58-62 

58-98 

59-26 

60-16 

0-35 

58-04 

58-78 

59-34 

59-81 

60-17 

60-47 

61-45 

0-40 

58-89 

59'58 

60-13 

60-60 

60-97 

61-30 

63-2 

0-45 

59-38 

60-06 

60-62 

61-07 

61-47 

61-83 

63-0 

0-50 

56-69 

-60-29 

60-84 

61-26 

61-72 

62-07 

63-3 

a 

c=0 

c=01 

0=0-2 

0=0-8 

o=0-4 

c=0-5 

c=l-0 

0-05 

31-3° 

33-68° 

35-46° 

36-36° 

37-22° 

38-0° 

39-9° 

o-io 

40-6 

42-4 

43-7 

44-64 

45-35 

45-92 

47-45 

0-15 

46-77 

48-20 

49-18 

49-93 

60-47 

50-92 

52-15 

0-20 

51-23 

52-27 

53-05 

63-64 

54-07 

54-42 

55-47 

0-25 

54-42 

55-22 

55-84 

56-31 

56-70 

57-01 

57-97 

0-30 

56-72 

57-38 

57-90 

58-30 

58-65 

58-94 

59-85 

0-35 

58-35 

58-94 

59-40 

59-79 

60-11 

60-38 

61>30 

0-40 

69-56 

60-09 

60-52 

60-89 

61-19 

61-46 

62-4 

0-45 

60-40 

60-89 

61-29 

61-67 

61-97 

62-24 

63-2 

0-50 

60-99 

61-43 

61-8 

62-2 

62-5 

62-8 

63-8 

HORIZONTAL   THRUST   OF   AN    ARCH. 


691 


THE  TABLES  OF  M.  GARIDEL. 
TABLE  III. 

Showing  the  Horizontal  Thrust  of  an  Arch,  the  Radiw  of  whose  Intrados 
is  Unity,  and  the  weight  of  each  Cubic  Foot  of  its  Material  and  that  of 
its  Loading,  Unity.  (See  Art.  344.) 

N.B.  To  find  the  horizontal  thrust  of  any  other  arch,  multiply  that  given 
in  the  table  by  the  square  of  the  radius  of  the  intrados  and  by  the  weight 
of  a  cubic  foot  of  the  material. 


=  0. 


a 

c=0 
P 

72- 

c=0-l 
P 

r2 

c=0-2 
P 

75- 

c=0-3 
P 
7e* 

c=0-4 
P 

7T 

c=0-5 
P 
r2 

0=1-0 

_P_ 
r* 

0-05 

0-08174 

0-14797 

0-21762 

0-28877 

0-36060 

0-43277 

0-79541 

o-io 

0-10279 

0-16370 

0-22588 

0-28862 

0-35164 

0-41481 

0-73161 

0-15 

0-11894 

0-17480 

0-23111 

0-28764 

0-34429 

0-40100 

0-68504 

0-20 

0-13073 

0-18191 

0-23322 

0-28460 

0-33603 

0-38747 

0-64488 

0-25 

0-13871 

0-18553 

0-23237 

0-27922 

0-32607 

037293 

0-60727 

0-30 

0-14333 

0-18604 

0-^*2874 

0-27145 

0-31416 

0-35687 

0-57041 

0-35 

0-14504 

0-18379 

0-22258 

0-26140 

0-30023 

0-33907 

0-53335 

0-40 

0-14422 

0-17913 

0-21415 

0-24924 

0-28437 

0-31953 

0-49560 

0-45 

0-14124 

0-17240 

0-20374 

0-23520 

0-26674 

0-29835 

0-45693 

0'50 

0-13649 

0-16396 

0-19168 

0-21957 

0-24760 

0-27573 

0-41728 

<  =  7°  30'. 


i 

c=0 
P 

r2 

c=0-l 
P 
"JST 

c=0-2 
P 
JS 

o=0-3 
P 
r2 

c=0-4 
P 

Ta" 

c=0-5 
P 

7T 

c  1-0 
P 

r2 

0-05 

0-06180 

0-12867 

0-19937 

0-27125 

0-34356 

0-41606 

0-77944 

o-io 

0-08514 

0-14666 

0-20930 

0-27237 

0-33561 

0-39895 

0-71618 

0-15 

0-10380 

0-16001 

0-21657 

0-27326 

0-33003 

0-38683 

0-67110 

0-20 

0-11813 

0-16948 

0-22089 

0-27237 

0-32384 

0-37533 

0-63286 

0-25 

0-12870 

0-17557 

0-22244 

0-26932 

0-31619 

0-36306 

0-59743 

030 

0-13598 

0-17866 

0-22134 

0-26403 

0-80673 

0-34943 

0-56'295 

0-35 

0-14040 

017909 

0-21783 

0-25661 

0-29542 

0-33424 

0-52846 

0-40 

0-14234 

0-17718 

0-21215 

0-24720 

0-28230 

0-31744 

0-49344 

0-45 

0-14211 

0-17323 

0  20454 

0-23598 

0-26751 

029910 

0-45763 

0-50 

0-14003 

0-16753 

0-19528 

0-22319 

0-25124 

0-27938 

0-42096 

1 

692 


APPENDIX. 


a 

c=0 
P 

I5r 

c=0-l 
P 

"ft" 

'  c=0-2 
P 

7*" 

c=0-8 
P 

r% 

c=0-4 
P 

r2 

c=0-5 
P 

7T 

c=l-0 
P 

r2 

0-05 

0-05310 

0-12265 

0-19488 

0-26748 

0-34018 

0-41293 

0-77681 

o-io 

0-07903 

0-14170 

0-20493 

0-26832 

0-33176 

0-39524 

0-71277 

0-15 

0  '09990 

0-15658 

0-21336 

0-27022 

0-32708 

0-38395 

0-66840 

0-20 

0-11631 

0-16781 

0-21931 

0-27083 

0-32234 

0-37386 

0-63145 

0-25 

0-12894 

0-17582 

0-22268 

0-26955 

0-31643 

0-36330 

0-59767 

0-30 

0-13835 

0-18096 

0-23361 

0-26627 

0-30895 

0-35163 

0-56510 

0-35 

0-14494 

0-18355 

0-22224 

0-26098 

0-29976 

0-33855 

0-53271 

0-40 

0-14905 

0-18384 

0-21878 

0-25380 

0-28888 

0-32399 

0-49995 

G'45 

0-15097 

0-18212 

0-21344 

0-24488 

0*27641 

0-30800 

046652 

0-50 

0-15099 

0-17860 

0-20642 

0-23439 

0-26247 

0-29065 

0-43232 

=  22°  30'. 


a 

c=0 
P 

7-2 

c=0-l 
P 

Te' 

c=0-2 
P 

r& 

c=0-3 
P 

r 

c=0-4 
P 

~r* 

c=0-5 
P 

r- 

c=l-0 
P 

7T 

0-05 

0-06102 

0-13346 

0-20621 

0-27899 

0-35178 

0-42458 

0-78857 

o-io 

0-08700 

0-15053 

0-21407 

0-27760 

0-34113 

0'4"466 

0-72233 

0-15 

0-10877 

0-16567 

0-22257 

0-27947 

0-33638 

0-39328 

0-67778 

0-20 

0-12635 

0-17785 

0-22936 

0-28087 

0-33239 

0-38391 

0-64150 

0-25 

0-14037 

0-18716 

0-23399 

0-28082 

0-32767 

0-37453 

0-60886 

0-30 

0-15129 

0-19381 

0-23640 

0-27902 

0-32166 

0-36432 

0-57773 

0-35 

0-15948 

0-19804 

0-23669 

0-27540 

0-31415 

0-35292 

0-54700 

0-40 

0-16525 

0-20005 

0-23497 

0-26999 

0-30506 

0-34017 

0-51608 

0-45 

0-16883 

0-20005 

0-23141 

0-26289 

0-29444 

0-32604 

0-48460 

0-50 

0-17047 

0-19824 

0-22617 

0-25423 

0-28238 

0-31060 

0-45241 

a 

o=0 
P 

~r% 

c=0-l 
P 

r2 

c=0-2 
P 

r 

c=0-8 
P 

75" 

c=0-4 
P 

rB 

c=0-5 
P 

r 

o  1-0 
•p 

72~ 

0-05 

0-09355 

0-16408 

0-23605 

0-30845 

0-38101 

0-45365 

0-81731 

o-io 

0-11297 

0-17592 

0-23922 

0-30263 

0-36609 

0-42957 

0-74711 

0-15 

0-13295 

0-18962 

0-24640 

0-30323 

0-36009 

0-41696 

0-70138  ' 

0-20 

0-15038 

0-20172 

0-25314 

0-30459 

0-35606 

0-40755 

0-66506 

0-25 

0-16493 

0-21160 

0-25834 

0-30513 

0-35193 

0-39876 

0-63299 

0-30 

0-17673 

0-21917 

0-26170 

0-S0427 

0-34688 

0-38951 

0-60282 

0-35 

0-18599 

0-22452 

0-26314 

0-30182 

0-34055 

0-37930 

0-57332 

0-40 

0-19293 

0-22777 

0-26271 

0-29773 

0-33280 

Q  -367  91 

0-54380 

0-45 

0-19774 

0-22906 

0-26050 

0-29202 

0-32361 

0-35524 

0-51385 

0-50 

0-20060 

0-22854 

0-25661 

0-28476 

0-31299 

0-34128 

0-48327 

HORIZONTAL   THRUST   OF   AN   ARCH. 

,=370  30'. 


693 


c=0 

c=0-l 

0=0-2 

c=0-3 

o=0-4 

c=0-5 

c=l-0 

a 

F 

P 

P 

P 

P 

P 

P 

re 

re 

re 

re 

T* 

r2 

re 

0-05 

0-14749 

0-21733 

0-28854 

0-36038 

t)-48255 

0-50490 

0-86784 

o-io 

0-15949 

0-22174 

0-28457 

0-34768 

0-41093 

0-47426 

0-79141 

0-15 

0-17605 

0-23233 

0-28886 

0-34553 

0-40226 

0-45904 

9-74322 

0-20 

0-19209 

0-24321 

0-29448 

0-34583 

0-39722 

0-44865 

0-70598 

0-25 

0-20627 

0-25282 

0-29948 

0-34619 

0-39294 

0-43972 

0-67382 

0-30 

0-21827 

0-26066 

0-30314 

0-34568 

0-38825 

0-43085 

0-64406 

0-35 

0-22805 

0-26659 

0-30521 

0-34388 

0-38259 

042133 

0-61529 

0-40 

0-23570 

0-27060 

0-30558 

0-34062 

0-37571 

0-41088 

0-58673 

0-45 

0-24130 

0-27275 

0-30427 

0-33586 

0-36749 

0-39916 

0-55787 

0-50 

0-24499 

0-27312 

0-30132 

0-32958 

0-35789 

0-38625 

0-52845 

45°. 


c=0 

c=0-l 

c=0.2 

c=0-8 

c=0-4 

c=0-5 

c=l-0 

a 

P 

P 

P 

P 

P 

P 

P 

re 

re 

? 

r2 

re 

r2 

r2 

0-05 

0-23105 

0-30081 

0-37162 

0-44305 

0-51485 

0-58688 

0-94881 

o-io 

0-23318 

0-29507 

0-35754 

0-42034 

0-48333 

0-54646 

0-86300 

0-15 

0-24478 

0-30079 

0-35708 

0-41355 

0-47013 

0-62678 

0-81059 

0-20 

0-25819 

0-30915 

0-36028 

0-41151 

0-46281 

0-51416 

0-77124 

0-25 

0-27104 

0-31752 

0-36410 

0-41074 

0-45744 

0-50417 

0-73809 

0-30 

0-28248 

0-32486 

0-36731 

0-40981 

045235 

0-49493 

0-70803 

0-35 

0-29216 

0-33073 

0-36935 

0-40803 

0-44674 

0-48547 

0-67939 

0-40 

0-29997 

0-33494 

0-36998 

0-40506 

0-44016 

0-47530 

0-65123 

0-46 

0-30589 

0-33745 

0-36907 

0-40072 

0-43240 

0-46412 

0-62294 

0-50 

0-30996 

0-33824 

0-36657 

0-39494 

0-42334 

0-45177 

0-59419 

1 

! 

691 


APPENDIX. 


TABLE  IY. 

Mechanical  Properties  of  the  Materials  of  Construction. 


2fote.—The  capitals  affixed  to  the  numbers  In 

B.  Barlow,  Report  to  the  Commissioners  of 

the  Navy,  &c. 
Be.      Be  van. 

Br.      Belidor,  Arch,  ffydr. 
Bru.    Brunei. 

C.  Couch. 

D.  W.  Daniell  and  Wheatstone,  Report  on  the 

Stone  for  the  Houses  of  Parliament. 

F.        Fairbairn. 

H.       Hodgkinson,  Report  to  the  British  As- 
sociation of  Science.  &c. 

K.        Kirwan. 


this  table  refer  to  the  following  authorities: 
La.  Lame. 

M.   Muschenbroek,  Inirod.  ad  Phil.  Nat.  i. 
Mi.  Mitis. 
Mt.  Mushct. 
Pa.  Colonel  Pasley. 
R.   Eondelet,  VArt  de  Eatir,  iv. 
Re.  Rennie,  Phil.  Trans.  &e. 
T.    Thomson,  Chemistry. 
Te.  Telford. 
Tr.  Tredgold,  Essay  on  the  Strength  oj 

Cast  Iron. 
W.  Watson. 


Names  of  Material!. 

Specific 
gravity. 

Weight  of  1 
cubic  foot 
in  Ibi. 

Tenacity 
per 

-TnT1 

Crushing 
force  p-r 
square  in,:h 
in  DM. 

Modulus  of 
elasticity  £. 

Modulu.1  of 
rupture  S. 

Acacia  (Eng.  growth) 

•710  B. 

4487 

16000  Be. 

1152000  B 

11202  B. 

Air  (atmospheric)  .    . 

•001228 

0-0768 

Alabaster  (yel.  Malta) 

2-699 

168-68 

Do.  (stained  brown) 

2-744 

171-50 

Do.  (oriental  white) 

2-780 

17062 

j  Alder  .... 

•800  M. 

50-00 

14186  M. 

6895  H. 

Antimony  (cast) 

4-500  M. 

281-25 

1066  M. 

Apple-tree 

•798  M. 

49-56 

19500  Be. 

Ash      .       .       .       j 

•690 
to   -845 

48-12 
58-81 

j-  17207  B. 

j  8688  H. 
\  9868  H. 

j-  1644800  B 

12156  B. 

Bay-tree     . 

•882  M. 

51-87 

12396 

7158  H. 

Bean  (Tonquin) 

1-080 

67-50 

2601600  B. 

20886  B. 

Beech                 .       J 

•854 

5887 

15784  B.' 

7733  H. 

'  1  Q^QAAA    "D 

OQQA    T? 

to   -690 

4312 

17S50  B. 

9863  dry  H. 

/•  loOoOUU    15. 

yooo  x>. 

Birch  (common) 

•792  B. 

4950 

15000       | 

4538  H. 
6402  dryH. 

j-  1562400  B. 

10920  B. 

Do.  (American) 

•648  B. 

40-50 

. 

11663  H. 

1257600  B. 

9624  B. 

Bismuth  (cast)    . 

9-810  M. 

61887 

'  3250  M. 

Bone  of  an  ox 

1-656  M. 

103-50 

5265 

Box  (dry)   . 

•960  B. 

6000 

19891  B. 

1C299  H. 

Brass  (cast) 

8-899 

52500 

17968  Re. 

10804  Re. 

8930000 

T>o.  (wire-drawn)    . 

8-544 

584-00 

Brick  (red) 

2-168  Re. 

135-50 

280 

807  Re. 

Do.  (pale  red) 

2-085  Re. 

130  81 

800 

562  Re. 

Brick-work 

1'800 

112'50 

Bullet-tree  (Berbice). 

1  C29  B. 

6431 

. 

. 

2610600  B. 

15686  B. 

Cane    .... 

0-400 

2500 

6800  Be. 

Cedar  (Canadian,fresh) 

0-909  0. 

5681 

11400  Be' 

5674  H. 

Do.  (seasoned) 

0-758 

47-06 

4912  H. 

Chalk  .       . 

2-784 
to  1-869 

17400 
116-81 

\-       '• 

884  Re. 

Chestnut     . 

0-657  R 

41  -06 

-lOOflA  T> 

Clay  (common)  . 

1-919  Br. 

119-98 

looUU  XV. 

Coal  (Welsh  furnace). 

1-887  Mt. 

88-56 

Do.  (coke)  . 

1-OOOMt 

62-50 

Do.  (Alfreton)  . 

1-285  Mt. 

77-18 

Do.  (Butterly)  . 

1  264  Mt. 

79-00 

Do.  (coke) 

1-100  Mt. 

6875 

• 

Do.  (Welsh  stone)    . 

1-368  Mt. 

85-50 

1 

Do.  (coke). 

1-890  Mt. 

8687 

Do.  (Welsh  slaty)     . 
Do.  (Derby,  cannel)  . 

1  409  Mt. 
1-278  Mt. 

88-06 
79-87 

Do.  (Kilkenny) 

1-6C2  Mt. 

100-12 

Do.  (coke). 

1-657  Mt. 

108-56 

Do.  (slaty). 

1-448  Mt. 

90-18 

I 

PROPERTIES   OF   MATERIALS   OF   CONSTRUCTION. 


695 


Tenacity 

Crushing 

Specific 

We  ght  of 

per 

force  per 

Modulu*  of 

Modulus  of 

Names  of  Material*. 

gravity. 

1  cubic  foo 
in  Ibs. 

square  inch 
in  Ibs. 

square  inch 
in  Ibs. 

elasticity  K. 

rupture  S. 

Coal  (Bonlavooneen)  . 
Do.  (coke)  . 

1-436  Mt 
1-596  Mt. 

89-75 
99-75 

Do.  (Corgee) 

1-403  Mt 

87-68 

Do.  (coke)  . 

1-656  Mt. 

108-50 

Do.  (Staffordshire)     . 
Do.  (Swansea)    . 

1-240 
1-357  K. 

78-12 
84-81 

Do.  (Wigan) 

1-268  K. 

79-25 

Do.  (Glasgow)    . 

1-290 

80-62 

Do.  (Newcastle) 

1-257  K. 

78-56 

Do.  (common  cannel) 

1-282  K. 

77-00 

Do.  (slaty  cannel) 

1-426  K. 

89-12 

Copper  (cast) 

8-607 

587-93 

19072 

Do.  (sheet) 

8-785 

549-06 

Do.  (wiredrawn) 

8-878 

560-00 

61228 

Do.  (in  bolts)   . 

48000 

Crab-tree    . 

0-765 

'47-80 

6499  H. 

Deal  (Christiana  mid- 

dle) . 

0-698  B. 

48-62 

12400 

1672000  B 

9864  B. 

Do.  (Memel  middle)  . 

0-590  B. 

86-87 

< 

1535200  B 

10886  B. 

Do.  (Norway  spruce) 
Do.  (English)     . 

0-340 
0-470 

21-25 
29-37 

'  17600 
7000 

Earth  (rammed) 

1-584  Pa. 

99-00 

Elder  .... 

0-695  M. 

48-43 

10230   * 

8467  H. 

Elm  (seasoned)  . 

0-588  C. 

86-75 

18489  M. 

10831.  H. 

699840,B, 

6078  B< 

Fir  (New  England)   . 

0-553  B. 

84-56 

. 

2191200-  B. 

6612  B. 

Do.  (Riga)  .        .        . 

0-753  B. 

47-06  1 

'  11549  B. 
to  12857  B. 

5748  H. 
o  688ft  H.. 

1328800  B. 
869600  B. 

6648  B. 
7572  B. 

Do.  (Mar  Forest) 

0-693  B. 

48-31 

Flint   .... 

2-630  T. 

164-87 

Glass  (plate) 

2453 

158-81 

9420 

Gravel 

1'920 

120-00 

Granite  (Aberdeen)  . 

2'526 

164-00 

Do.  (Cornish)  . 

2-662 

16680 

Do.  (red  Egyptian)  . 

2-654 

165-80 

Hawthorn  . 

0-91    Be. 

88-12 

10500  Be. 

Hazel  .... 

0-86    Be. 

53-75 

18000  Be. 

Holly  .... 

0-76    Be. 

47-5 

16000  Be. 

Horn  of  an  ox     . 

1-689  M. 

105-56 

8949 

Hornbeam  (dry) 

0-760  R. 

47-50 

20240  Be. 

7289  H. 

Iron  (wrought  Eng.)  . 

7-700 

481-20 

25itons,  La. 

Do.  (in  bars) 

j    -7600 
1  to7-800 

475-50 
487-00 

25itons,  La. 

Do.  (hammered) 

30  tons,  Bru. 

Do  (Russian)  in  bars 

m 

27  tons,  La. 

Do  (Swedish)  in  bars 

. 

12  tons,  R. 

Do.  (English)  in  wfre 

l 

j  36  to  43 

l-10th  inch  diameter 

1  * 

• 

;     tons,  Te. 

Do.  (Russian)  in  wire 
l-20th  to  1-SOth  inch 
diameter  . 

| 
1 

.      • 

I  60  to  91 
tons,  La. 

Do.  rolled  in  sheets 

and  cut  lengthwise  . 
Do.  cut  crosswise 

• 

•      • 

14  tons,  Mi. 
18  tons,  Mi. 

Do.    in    chains,    oval 

links,  6  inches  clear, 

iron  ^  inch  diam.  . 

4 

2Htons,Br. 

Do.  (Brunton's)  with 

stay  across  link 
Iron,  cast  (old  Park) 

• 

•      • 

25  tons,  B. 

18014400  T. 

48240  T 

Do.  (Adelphi) 

18858600  T. 

5860  T. 

Do  (Alfreton) 

17686400  T. 

4046  T. 

Do.  (scrap) 

18082000  T. 

5828  T! 

Do.    (Carron,    No.   2 

cold  blast) 
Do.  (hot  blast)  . 

7-066  H. 
7-046  H. 

441-62 
440-87 

16683  H. 
18505  II. 

06875  H. 
08540  H. 

17270500  H. 
16085000  H. 

J8556  H  .*! 
7508  H.*! 

•  The  numbers  marked  thus  *  are  calculated  from  the  experiments  of  Messrs.  Hodgkinson  &  Fairbairn, 


600 


APPENDIX. 


1 

N*met  of  Material.. 

Specific 
gravity. 

Weight  of  1 
cubic  foot 
in  Ibs. 

Tenacity 
per 
square  inch 
in  Ib.. 

Crushing 
force  per 
«)uare  inch 
in  Ib, 

Modulus  of 
elasticity  E. 

Modunii  of 
rupture  8. 

Iron,  cast  (do.  No.  8, 

Carron  cold  blast    . 

7-094  F. 

443-37 

14200  H. 

115442  H. 

16246966  F. 

35980  F.* 

Do.  (hot  blast)    . 

7-056  F. 

441-00 

17755  H. 

183440  H. 

17873100  F. 

42120  F.* 

Do.    Devon.    No.    8, 

cold  blast) 

7-295  H. 

455-93 

22907700  H. 

86288  H.* 

Do.  (hot  blast)   . 

7-229  H. 

451-81 

29107  H. 

145485  H. 

22473650  H. 

43497  H." 

Do    (Buflery,  No.  1, 

cold  blast) 

7-079  H. 

442-48 

17466  H. 

93366  H. 

15381200  H 

37503  H.* 

Do.  (hot  blast)  . 

6  998  H. 

487-37 

18434  H. 

86397  H. 

18730500  H. 

85316  H.* 

Do.  (Coed  Talon,  No. 

2,  cold  blast)   . 

6-955  F. 

484-06 

18855  H. 

81770  H. 

14813500  F. 

33104  F  * 

Do.  (hot  blast)   . 

6-968  F. 

435-50 

16676  H. 

82739  H. 

14822500  F. 

83145  F.* 

Do.  (Coed  Talon,  No. 

8,  cold  blast)    . 

7-I94F. 

449-62 

. 

17102000  F. 

43541  F* 

Do.  (hot  blast)    . 

6-970  F. 

435-62 

> 

t 

14707900  F. 

40159  F.* 

Do.    (Elsicar,  No.  1, 

cold  blast)       .        . 

7-080  F. 

489-87 

13981000  F. 

84862  F.* 

Do     (Milton,  No.   1, 

hot  blast). 

.  6-976  F. 

486-00 

t 

t 

11974500  F. 

28552  F.* 

Do.  (Muirkirk,  No.  1, 

cold  blast) 

7-118  F. 

444-56 

t 

t 

14003550  F. 

85923  F.* 

Do.  (hot  blast)   . 

6-953  F. 

434-56 

m 

13294400  F. 

33850  F.* 

Ivory  .... 

1-826  P. 

114-12 

16-626 

Laburnum  . 

0-92   Be. 

57-50 

10500  Be. 

Lance-wood 

1-022 

63-87 

24696 

Larch  .... 

0-522  B. 

32-62 

10220  E. 

8201  H. 

(green) 

897600  B. 

4992  B. 

Do.  (second  specimen) 

0-560  B. 

85-00 

8900  B. 

5568  H. 
(dry) 

1052800  B. 

6894  B. 

Lead  (cast  English)    . 

11-446  M. 

717-45 

1824  Ee. 

V>"V  ) 

720000  Tr. 

Do.  (milled  sheet)     . 

11-407  T. 

712-93 

3328  Tr. 

Do.  (wire)  . 

11-317  T. 

705-12 

2581  M. 

Lignum  vitae       [ous) 

1-220 

76-25 

11800  M. 

Limestone    (arenace- 

2-742 

171-37 

Do.  (foliated)  . 

2-887 

177-81 

Do.  (white  fluor)    . 

8-156 

197-25 

Do.  (green)     . 

3-182 

198-87 

Lime-tree   . 

0-760 

47-50 

23500  Be. 

Lime  (quick) 

0-488  Br. 

52-68 

Mahogany  (Spanish)  . 
Maple  (Norway) 

0-800 
0-793 

50-00 
49-56 

16500 
10584 

8198  H. 

Marble  (white  Italian) 
Do.  (black  Galway)  . 
Mercury  (at  82°) 

2-638  H. 
2-695  H. 
13-619 

16487 
168-25 
851-18 

. 

2520000  T. 

1062 
2664 

Do.  (at  60°)     . 

13-580 

848-75 

Marl    .       .               -j 

1-600 

100-00 

to  2-877  T. 

118-81 

Mortar        .       .       ' 

1  751  Br. 

107-18 

50 

Oak  (English)     . 

0-934  B. 

58-87 

17-800  M.-! 

4684  H. 
9509  H. 

[•  1451200  B. 

0082  B. 

Do.  (Canadian). 

0-872  B 

54-50 

I 

1  0-253  M.j 

(dry) 
4231  H. 
9509  H. 

Ul48800B. 

0596  B. 

Do.  (Dantzic)     . 
Do.  (Adriatic)    . 
Do.  (African  middle). 

0-756  B. 
0-993  B. 
0-972  B. 

47-24 
62-06 
6075 

12-780 

(dry) 

1191200  B. 
974400  B. 
2283200  B. 

8742  B. 
8298  B. 
18566  B. 

Pear-tree    . 
Pine  (pitch. 

0-661  M. 
0-660  B. 

41-81 
4125 

'7818M 

7518  H. 

1225600  B. 

9792  B. 

Do.  (red)    .        .        ] 
Do.  (Amer.  yellow)  . 
Plane-tree  . 

0-657  B. 
0-461  C. 
0-64   Be. 

4106 

28-81 
40-00 

11700  Be. 

5375  H.   ' 
5445  H. 

1840000  B. 
1600000  Tr. 

8946  B. 

Plum-tree  . 

0-785  M. 

49-06 

11-851- 

9367  H. 
8657  H. 

(wet) 

Poplar.       .       .       . 

0-388  M. 

23-98 

7200  Be.- 

8107  H. 
5124  H. 

Pozzolano   . 

2-677  K. 

169-87 

(dry) 

Sand  (river) 

1*886 

117-R7 

Serpentine  (green) 

2-574  K. 

11  i  Ol 

168-87 

PROPERTIES   OF   MATERIALS    OF   CONSTRUCTION. 


097 


1 

Names  of  mat.  r  al>. 

Specific 
grav.ty. 

Weight  of 
1  cubic  foot 

in  iha. 

Tenacity 

^JTU 

Crushing 
/ua're'hich 

Modulu.  of 
ehuticity  E. 

Modnlu.  of 
rupture  8. 

Shingle    -           -           -    1  424  Pa. 

89-00 

Silver  (standard)           -    10'812  T. 

644-50 

40902  M. 

Slate  (Welsh)      -           -    2'888 
do.  (Westmoreland)    -    2-791  W. 

180-50 

12800 

- 

15800000  Tr 

1  OQAAAAA    m_ 

11766  Re. 

do.  (Valontia)  -           -    2-880  Re'. 

180*00 

iiyuvuUu  AJ 

KOOA  "PA 

9600 

1r»7fWWW»A   T 

0/^0  IwO. 

Steel  (soft)          -           -    7-780 

486-25 

120000 

UOOOO  Tr 

do.   (razor-tempered)  -    7-840 
Stone  (Ancaster)            -    2'182  D.W 

490-00 
186-37 

150000 

29000000  T. 

do.    Barnack   -           -    2*090  D.W 

18062 

do.    Binnie      -           -    2-194  D.W 

187*12 

do.    Bolsover  -           -    2-316  D.W 

144-75 

do.    Box           -           -    1-889  D.W 

114-98 

do.     Bramham  Moor  -    2-008  D.W 

125-50 

do.    Brodsworth         -    2-093  D.W 

130*81 

do.    Cadeby     -           -    1-951  D.W 
do.    Oslthncss  -           -   *2*764  R<? 

121-98 

do.'    Craigleith-           -    2-266  D.W 

141-62 

5142  Ee. 

do.     Chilmark  (A)        -    2-366  D.W 

147-87 

do.    Chilmark  (B)       -    2-383  D.W 

148-98 

do.     Chilmark  (C)       -    2-481  D.W 

155-06 

do.    Darby  Dale   (Stan-! 

cliffe)            -           -    2-626  D.W 

164-25 

do.    Giffneuk  -           -    2  280  D.W 

189-37 

do.    Gunbarrel  (Stanley)  2  260  D.W 

141-25 

do.    Ham  Hill-           -   ,2260  D.W 

14125 

do.    Haydon    -           -   '  2*040  D  W 

127-50 

do.    Heddon    -           -   J2'229  D.W 

139-81 

do.    Hildenly  - 

2-098  D.W 

131-12 

do.    Hookstone 

2-253  D.W 

14081 

do.    Huddlestone 

2  147  D.W 

134*18 

do.    Little  Hulton 

2'857  H. 

-J4.IT.0-J 

774  TT 

do.    Keuton    - 

2-247  D!W 

J.4(  oi 
,  140-43 

<  I3k   -H. 

do.    Ketton 

2-045  D.W 

127-81 

do.    Ketton  rag 

2-490  D.W. 

155-62 

do.    Mansfield,  or  Lind- 

ley's  red 

2-838  D.W. 

14612 

do.    white 

2-277  D.W. 

14281 

do.    Morley  Moor 

2-053  D.W. 

128-81 

do.    Park  Nook 

2-138  D.W. 

183-62 

do.    Park  Spring 

2-321  D.W. 

145-06 

do     Portland  (Waycroft 

Quarry 
do.    Redgate    - 

2-145  D.W. 
2  239  D.W. 

184-06 
189-93 

do.    Roach  Abbey 

2-134  D.W. 

O.K771   TT 

183-87 

2868  H. 

do.    Stanley     - 

a  0(  1    II. 

2-227  D.W. 

161  "06 
189-18 

do.    Taynton  - 

2-108  D.W. 

181-48 

do.    Totterwhoe 

2-891  D.W. 

118-18 

9 

do.    Jackdaw  crag 

2-070  D.W. 

O.QOA  TT 

129-37 

1116  H 

do.     Yorkshire  nag 
do.    Green  Moor 

z  oM  n.. 
2-584  Re. 

140  "00 
158-37 

. 

. 

. 

2010  Re. 

Sycamore 

0-69   Be. 

43-12 

18000  Be. 

'eak  (dry) 

0*657  C. 

41-06 

15000  B. 

12101  H. 

2414400  B. 

4772  B. 

Mle  (common)    - 
Tin  (cast) 

1-815  Br. 
7*291  Tr. 

113-48 
455-68 

6822  M. 

- 

4608000  Tr. 

Water  (sea) 

1027T. 

64-18 

do.    (rain) 

1-000 

62-50 

Walnut    - 

0-671  M. 

41-93 

8130  M. 

6645  H. 

7667 

„                    . 

820000  TT. 

Willow  (dry)      - 

0-890 

24-87 

14000  Be. 

Yew  (Spanish)    - 

0-807  M. 

50-48 

8000  Be. 

Zinc 

7-028  W. 

48925 

- 

- 

8680000  Tr. 

Note.— The  experiments  of  Mr.  Hodgkinson,  from  which  the  moduli  of  rupture  of  stones  contained 
in  the  last  column  of  the  above  table  have  been  calculated,  together  with  those  detailed  in  Art.  410., 
i.j.  in  part  of  a  more  extended  inquiry,  which,  when  completed,  will  be  laid  before  the  Royal  Society. 
Tin-  crushing  forces  given  in  the  above  tables  are  in  every  case  determined  by  the  compression  of 
j>;  i.-ms  of  such  a  height  as  to  allow  the  rupture  to  take  place  by  the  sliding  of  the  upper  portion  freely 
olf.  along  its  plane  of  separation  from  the  lower  (see  Art.  406).  The  experiments  of  Mr.  Hodgkinson 
have  shown  that  all  results  obtained  without  reference  to  this  circumstance  are  erroneous.  (Se« 
^castrations  of  Mechanics,  p.  402.) 


APPENDIX. 


TABLE  V. 

Useful  Numbers. 


it    . 

Log.* 

Log.£rt 

1 

ft 

x*  .    . 

1_ 

*'_' 

4/* 

4^ 

4/2 


=3-1415927 
=0-4971499 
:1-1447299 

=0-3183099 
=9-8696044 
=0-1013212 
=1-7724538 
=0-5641896 
:1-4142136 


1 

V5 


71 


Vi 
V! 


=0-7071068 
=4-4428829 
=2-2214415 
=0-4501582 
=1-2533141 
=0-7978846 


£      .  =2-7182818 

Log.  c •      .        .        .  =0-4342945 

Modulus  of  common  logarithms =-434294482 

Log.  of  ditto =9-6377843 

g =32-19084 

4/0 =5-67363 

Log.  g =1-5077222 

Inches  in  a  French  m^tre =39*37079 

Log.  of  ditto =1-5951741 

Feet  in  ditto =3-2808992 

Log.  of  ditto =0-5159929 

Square  feet  in  the  square  metre =10*764297 

Acres  in  the  Are =0-024711 

Lbs.  in  a  kilogramme =2-20548 

Log.  of  ditto =0-3435031 

Imperial  gallons  in  a  litre =0-2200967 

Lbs.  per  square  inch  in  1  kilogramme  per  square  millimetre  =1422 

Owts.  ditto,  ditto =12'7 

Volume  of  a  sphere  whose  diameter  is  1          ...  =0-5235988 

Arc  of  1°  to  rad.  1 =0-017453293 

Arc  of  1'  to  rad.  1 =0-000290888 

Arc  of  1"  to  rad.  1 =0-000004848 

Degree  in  an  arc  whose  length  is  1 =57-295780° 

Grains  in  1  oz.  avoirdupois =437i 


USEFUL   NUMBERS.  699 

Grains  in  1  Ib.  ditto =7000 

Grains  in  a  cubic  inch  of  distilled  water,  Bar.  30  in.,  Th.  62°  =252-458 

Cubic  inches  in  an  ounce  of  water =1*73298 

Cubic  inches  in  the  imperial  gallon =277'276 

Feet  in  a  geographical  mile =6075*6 

Log.  of  ditto =3-7835892 

Feet  in  a  statute  mile =5280 

Log.  of  ditto =3-7226339 

Length  of  seconds'  pendulum  in  inches    .....  =39-19084 

Cubic  inches  in  1  cwt.  of  cast  iron =430-25 

Bar  iron  .         .        .        .        .  =397'60      . 

Cast  brass         ....  =368-88 

Cast  copper      ....  =352'41 

Cast  lead =272-80 

Cubic  feet  in  1  ton  of  paving  stone  .....  =14-835 

Granite =13'505 

Marble =13-070 

Chalk =12-874 

—  Limestone      .....  =11-273 
Elm =64-460 

—  Honduras  mahogany       .        .        .  =64-000 
Mar  Forest  fir        ....  =51-650 

Beech =51-494 

Kigafir =47'762 

—  Ash  and  Dantzic  oak      .        .        .  =47'158 

—  Spanish  mahogany         .        .        .  =42*066 
English  oak =36-205 

To  find  the  weight  in  Ibs.  of  1  foot  of  common  rope,  multi- 
ply the  square  of  its  circumference  in  inches  by         .        *044  to  -04C 
Ditto  for  a  cable -027 

Note. — The  numerical  values  of  the  function  of  *  in  this  table  were  calcu- 
lated by  Mr.  Goodwin.  These,  together  with  the  numbers  of  cubic  inches  and 
feet  per  cwt.  or  ton  of  different  materials,  are  taken  from  the  late  Dr.  Gregory's 
excellent  treatise,  entitled  Mechanics  for  Practical  Men.  The  other  numbers 
of  the  table  are  principally  taken  from  Mr.  Babbage's  Tables  of  Logarithm* 
and  the  Aide  Memoire  of  M.  Morin. 


THE  END. 


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